# Hybrid Neural Models For Sequence Modelling: The Best Of Three Worlds

Marco Dinarelli1 Loïc Grobol2,3
(1) LIG, Bâtiment IMAG - 700 avenue Centrale - Domaine Universitaire de Saint-
Martin-d’Hères
(2) Lattice CNRS, 1 rue Maurice Arnoux, 92120 Montrouge, France
(3) ALMAnaCH Inria, 2 rue Simone Iff, 75589 Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>

We propose a neural architecture with the main characteristics of the most
successful neural models of the last years: bidirectional RNNs, _encoder-
decoder_ , and the _Transformer_ model. Evaluation on three sequence labelling
tasks yields results that are close to the state-of-the-art for all tasks and
better than it for some of them, showing the pertinence of this hybrid
architecture for this kind of tasks.

KEYWORDS: Neural Networks, sequence modelling, MEDIA, WSJ, TIGER

##  1 Introduction

Sequence labelling is an important NLP task, many NLP problems can be modeled
indeed as sequence labelling. The most traditional labelling problems are
Part-Of-Speech (POS) tagging, syntactic chunking, Named Entity Recognition
(NER) Collobert et al., (2011), and also Spoken Language Understanding (SLU)
in the context of human-machine dialog systems De Mori et al., (2008).

Moreover there are other NLP problems that can be decomposed into several
processing steps, the first of which can be often modeled as a sequence
labelling problem.

We can place among such problems the syntactic constituency parsing, which can
be decomposed into POS tagging and constituents analysis Collins, (1997) ; the
coreference chain resolution Soon et al., (2001); Ng & Cardie, (2002),
composed of mention recognition from raw text and coreferent mention detection
; and also the structured named entity recognition Grouin et al., (2011);
Dinarelli & Rosset, (2012a, b).

More widely, machine translation and syntactic constituency parsing can be
modeled as end-to-end sequence prediction problems Sutskever et al., (2014);
Bahdanau et al., (2014); Vaswani et al., (2017); Vinyals et al., (2015), as
well as a wide class of language understanding tasks Devlin et al., (2018). It
would be possible thus, at least in principle, to design a model for sequence
prediction in a multi-task learning framework, in order to deal with most of
the NLP tasks, which shows in turn the interest in the research for sequence
modelling problems.

The NLP problems that can be decomposed into several processing steps can be
solved also with a single model. For example the syntactic constituency
parsing can be performed end-to-end with a model performing POS tagging and
constituent analysis at the same time, e.g. Rush et al., (2012). However, even
in such cases, pre-training representations with neural networks and fine-
tuning them afterwards on simpler tasks, like sequence labelling, can still
improve the prediction performance Peters et al., (2018). We can also consider
this training procedure similar to embedding pre-training, which effectiveness
has already beed proved Lample et al., (2016); Ma & Hovy, (2016).

In this paper we focus on neural architectures for sequence modelling in the
sense this is meant in problems such as POS tagging, morpho-syntactic
analysis, and semantic chunking and tagging. The latter is performed for SLU
tasks in the context of human-machine dialog systems De Mori et al., (2008).

We inspired our work from Chen et al., (2018), from which we also inspired the
title of our paper, and which proposes neural hybrid models combining the
_Encoder-Decoder_ (or _Sequence-to-Sequence_) and _Transformer_ architectures.
We propose an architecture with the main characteristics of the most
successful neural models of the last few years: the bidirectional RNNs Lample
et al., (2016); Ma & Hovy, (2016), the _Sequence-to-Sequence_ model Sutskever
et al., (2014); Bahdanau et al., (2014), and the _Transformer_ Vaswani et al.,
(2017).

We evaluate our architecture on three traditional tasks of sequence labelling:
Spoken Language Understanding (SLU) in French (MEDIA corpus) Bonneau-Maynard
et al., (2006), POS tagging in English (WSJ corpus) Marcus et al., (1993), and
morpho-syntactic tagging in German (TIGER corpus) Brants et al., (2004). The
results outperform often the state-of-the-art, and allow in any case to
conclude that our architecture finds its place among neural architectures.

##  2 Neural Architectures

The proposed neural models are inspired from the _LSTM+CRF_ architectures
Lample et al., (2016); Ma & Hovy, (2016) in the way input encoding (tokens,
characters, etc.) is performed. They are inspired from the _Sequence-to-
Sequence_ architecture for the overall architecture, and from the models
proposed in our previous work Dinarelli & Tellier, (2016a, b); Dinarelli et
al., (2017); Dupont et al., (2017); Dinarelli & Grobol, (2019) for making
predictions based on a bidirectional context on the output side (labels).
Afterwards we added to this architecture some of the characteristics of the
_Transformer_ model.

This work constitutes a step ahead with respect to Dinarelli & Grobol, (2019).
While we have not run full experiments to show the improvements with respect
to our previous work, and results are equivalent in some cases, some outcomes
suggest that the model proposed here is more effective. In any case the model
described here is more general, as the architecture has been slightly modified
so that it can be applied also to tasks like machine translation and syntactic
parsing, though we do not evaluate on these tasks in this work. Moreover in
other cases results are state-of-the-art, though they are obtained on data
(TIGER) on which we did not evaluate previously.

###  2.1 Encoder

The encoder of our network is made of a bidirectional GRU Cho et al., (2014),
which takes as input a sequence $S^{lex}$ representing input tokens as the
concatenation of word embeddings $\left(E_{w}(w_{i})\right)$ and
representations computed from their characters $h_{c}(w_{i})$. The latter is
computed by an _ad-hoc_ recurrent layer, in a fashion similar to those used in
the literature Ma & Hovy, (2016), with the difference that we use a GRU
recurrent layer instead of a convolution layer.

The character-level representation of a word $w$ is computed like in Dinarelli
& Grobol, (2019):

$\displaystyle S^{c}(w)$ $\displaystyle=(E_{c}(c_{w,1}),\dots,E_{c}(c_{w,n}))$
(1) $\displaystyle(h_{c}(c_{w,1}),\dots,h_{c}(c_{w,n}))$
$\displaystyle=\operatorname{GRU}_{c}(S^{c}(w),h_{c}^{0})$ $\displaystyle
h_{c}(w)$
$\displaystyle=\operatorname{FFNN}(Sum(h_{c}(c_{w,1}),\dots,h_{c}(c_{w,n})))$

where $E_{c}$ is the matrix of character embeddings, $c_{w,i}$ is the _i-th_
character of the word $w$, $S^{c}(w)$ is the sequence of character embeddings
for the word $w$, $\operatorname{GRU}_{c}$ is the GRU layer for characters,
$h_{c}(c_{w,i})$ is the hidden state associated to the _i-th_ character of the
word $w$. $\operatorname{GRU}_{c}$, like $\operatorname{GRU}_{w}$, is a
bidirectional GRU layer.

The full representation of the _i-th_ word $w_{i}$ in a sequence is computed
as:

$\displaystyle S^{lex}$
$\displaystyle=([E_{w}(w_{1}),h_{c}(w_{1})],\dots,[E_{w}(w_{N}),h_{c}(w_{N})])$
(2) $\displaystyle(h_{w_{1}},\dots,h_{w_{N}})$
$\displaystyle=\operatorname{GRU}_{w}(S^{lex},h_{w}^{0})$

Since $\operatorname{GRU}_{w}$ scans the sequence forward and backward,
$h_{w_{i}}$ depends on the word $w_{i}$ and its context. When additional
features are available at token level, they are embedded in the same way as
words and concatenated to word embeddings in the sequence $S^{lex}$.

###  2.2 Decoders

Our model uses a representation of both left and right context at label level
like proposed in Dinarelli & Tellier, (2016a); Dinarelli et al., (2017);
Dupont et al., (2017); Dinarelli & Grobol, (2019).

We use a $\operatorname{\overleftarrow{\operatorname{GRU}}}_{e}$ _backward_
layer for encoding the right context, and a
$\operatorname{\overrightarrow{\operatorname{GRU}}}_{e}$ _forward_ layer for
the left context with respect to a given labelling position $i$ in the input
sequence. These layers take as input the lexical level representation computed
by the encoder and the label embeddings $E_{e}(e_{i})$, which makes them
similar to the decoder used in the original _Sequence-to-Sequence_
architecture for machine translation Sutskever et al., (2014); Bahdanau et
al., (2014). An evolution with respect to such architecture is the use of two
decoders, one for the left label-level context and one for the right label-
level context. This idea of using both left and right context at the output
item’s level has attracted some interest in the last few years, and has been
applied in different variants to machine translation Xia et al., (2017); Zhang
et al., (2018a).

The right label-level context is computed by the _backward_
$\operatorname{\overleftarrow{\operatorname{GRU}}}_{e}$ decoder as follows:

$\overleftarrow{h_{e_{i}}}=\operatorname{\overleftarrow{\operatorname{GRU}}}_{e}([h_{w_{i}},E_{e}(e_{i+1})],\overleftarrow{h_{e_{i+1}}})$
(3)

The left label-level context $\overrightarrow{h_{e_{i}}}$ is computed in a
similar way by the _forward_
$\operatorname{\overrightarrow{\operatorname{GRU}}}_{e}$ decoder.

###  2.3 Output Layer

In order to allow the model to make global decisions when predicting the
current label, we add an output layer on top of the backward decoder, made of
an affine transformation followed by a _log-softmax_ activation. This computes
log-probabilities of backward predictions:

$\displaystyle\operatorname{log-P}(\overleftarrow{e_{i}})$
$\displaystyle=\operatorname{log-
softmax}(W_{bw}[h_{w_{i}},\overleftarrow{h_{e_{i}}}]+b_{bw})$ (4)
$\displaystyle\overleftarrow{e_{i}}$
$\displaystyle=\operatorname{argmax}(\operatorname{log-P}(\overleftarrow{e_{i}}))$

The forward decoder predicts labels using as input the label-level context
representations $\overrightarrow{h_{e_{i}}}$ and $\overleftarrow{h_{e_{i}}}$,
as well as the lexical level information $h_{w_{i}}$ computed by the encoder:

$\displaystyle\operatorname{log-P}(\overrightarrow{e_{i}})$
$\displaystyle=\operatorname{log-
softmax}(W_{o}[\overrightarrow{h_{e_{i}}},h_{w_{i}},\overleftarrow{h_{e_{i}}}]+b_{o})$
(5) $\displaystyle\overrightarrow{e_{i}}$
$\displaystyle=\operatorname{argmax}(\operatorname{log-P}(\overrightarrow{e_{i}}))$

In order to strengthen the global character of the decision, the final output
of the model is computed as the arithmetic mean (in the log space, which gives
a geometric mean of probabilities) of the _forward_ and _backward_ log-
probabilities:
$\frac{1}{2}(\operatorname{log-P}(\overrightarrow{e_{i}})+\operatorname{log-P}(\overleftarrow{e_{i}}))$.
This _teaches_ the model to compute real good predictions also in the backward
phase, instead of basing decisions on raw heuristics, or even on the forward
(left) context only. Indeed, the backward decoder has only a partial view of
the output level information, while the forward decoder uses both left and
right context.

###  2.4 Learning

All models described in this paper are learned minimizing the minus log-
likelihood $\mathcal{LL}$ of the training data. More precisely:

$-\mathcal{LL}(\Theta|D)=-\sum_{d=1}^{|D|}\sum_{i=1}^{N_{d}}\frac{1}{2}(\operatorname{log-P}(\overrightarrow{e_{i}})+\operatorname{log-P}(\overleftarrow{e_{i}}))+\frac{\lambda}{2}\left|\Theta\right|^{2})$
(6)

The first sum scans the data $D$ of size $|D|$, while the second sum is over
training examples $S_{d}$, of size $N_{d}$.

Since data used in this work for evaluation are relatively small, while our
models are relatively complex, we add a Gaussian regularization term $L_{2}$
with a coefficient $\lambda$.

###  2.5 _The best of three worlds_

The neural architecture described so far uses the same ideas introduced in
Dinarelli & Tellier, (2016a); Dinarelli et al., (2017); Dupont et al., (2017);
Dinarelli & Grobol, (2019) for predicting labels using both representations of
left (forward) and right (backward) contexts, and for both input-level
information (words, characters, etc.) and labels. Our model integrates also
the characteristics of bidirectional RNNs and of Encoder-Decoder
architectures. Moreover we use two decoders, instead of one like in the
original architecture.

Starting from this neural architecture we added also some of the
characteristics of the _Transformer_ model.

We note that the paper from which we inspired our work Chen et al., (2018)
investigated the combination of two Encoder-Decoder architectures, one based
on recurrent layers Sutskever et al., (2014); Bahdanau et al., (2014), while
the other is based on the _Transformer_ model, which is not a recurrent
network. This work shows that such combination is possible and effective
compared to the two combined architectures.

In this work we combined three different architectures with the goal of
putting together their strengths, and trying to overcome some limitations.

First of all, in the tasks of sequence labelling used for evaluation in this
work, there is a _one-to-one_ correspondence between input and output items
(words and labels, respectively). Our preliminary experiments on mention
detection for coreference resolution using a _Transformer_ , as well as some
experiments in the literature Guo et al., (2019), suggest that ignoring such
correspondence leads to a remarkable performance drop.111F1 measure drops more
than 3 points using the _Transformer_ in our experiments For this reason the
encoder and the decoders used in our architecture exploit the one-to-one
correspondence of sequence labelling tasks. We are aware though this
information could not be used in _real_ sequence-to-sequence problems such
like machine translation or syntactic parsing.

Second, we chose an Encoder-Decoder overall architecture since this proved to
be the most effective for sequence-to-sequence problems (including sequence
labelling). Moreover, we showed that keeping a label-level context into
account, in some sequence labelling problems, using a decoder is more
effective than doing the same with a CRF neural layer Dinarelli et al.,
(2017), we will show the same also in this work. As we already mentioned, in
our architecture we use two decoders. This solution allows for an even more
effective modelling of a label-level context. This solution is not possible,
at least in principle, with the original _Transformer_ model.

Third, we decided to integrate some of the characteristics of the
_Transformer_ in order to overcome limitations of RNNs concerning the learning
phase. As discussed in Vaswani et al., (2017), RNNs are difficult to train
because of the length of paths the learning signal has to back-propagate
through. The length of such paths is the consequence of the recurrent nature
of the networks.

The _Transformer_ model Vaswani et al., (2017) proposed an alternative, non-
recurrent Encoder-Decoder architecture based on a multi-head attention
mechanism. Recent works Dehghani et al., (2018); Dai et al., (2019) suggest
however that using some kind of recurrent connections _Transformer_ ’s
performance can be improved.

Another interesting characteristic of the _Transformer_ is the use of residual
connections, or skip-connections Bengio et al., (2003). This characteristic
allows for alleviating the vanishing gradient problem of deep networks.
Indeed, the use of gates in recurrent networks reduces the amplitude of this
problem, however the literature and our own experiments suggest that for long
sequences, gates are not sufficient for an effective learning.

The attention mechanism does not suffer from this problem. Each element of the
input sequence is related to a small, fixed number of layers in the attention
mechanism. The back-propagation path of the learning signal is thus much
shorter than that of RNNs. Moreover the use of skip-connections contributes in
keeping a higher learning signal amplitude, allowing to skip a whole layer,
over which the learning signal would be weakened (because of gradient
multiplications).

We can interpret each block of a _Transformer_ architecture, both encoder and
decoder, as the combination of a sub-module which _“encodes”_ contextual
features, we call this sub-module _ctx-encoder_ , and a _feed-forward_ sub-
module which _“maps”_ contextual features into a deeper feature space, we call
the second sub-module _deep-encoder_. The output of the two sub-modules is
added to their respective input (skip-connection) and normalized with a _layer
normalization_. This interpretation is shown in the figure 1 in the middle.
Skip-connections are highlighted in red.

outputoutputoutput _Add &Norm_ _Add &Norm_ _Add &Norm_ _deep-encoder_ _FFNN_
_FFNN_ _Add &Norm_ _Add &Norm_ _Add &Norm_ _ctx-encoder_ _Multi-head
Attention_ _GRU_ inputinputinput Figure 1: High-level architecture, from a
conceptual point-of-view, of each block of a _Transformer_ model (in the
middle). On the left we show its instantiation in the _Transformer_ , on the
right we show its instantiation in our architecture, where we use a
bidirectional GRU layer to encode contextual features

The _Transformer_ proposed in Vaswani et al., (2017) instantiate the generic
architecture described above using as _ctx-encoder_ the multi-head attention
mechanism, while it uses a two-layer feed-forward network as _deep-encoder_. A
single block of this architecture is shown in the figure 1 on the left.

We modified our neural architecture for instantiating the generic
architecture. In our solution we use a bidirectional GRU layer as _ctx-
encoder_ , while we use exactly the same sub-module as the _Transformer_ as
_deep-encoder_. A block of our architecture is shown in the figure 1 on the
right.

The characteristics of the _Transformer_ model we integrated in our
architecture are thus the skip-connections and layer normalization (_Add
&Norm_ in the figure 1), as well as the two-layer feed-forward sub-module to
re-encode contextual features once the skip-connection has been applied
(_FFNN_ in the figure 1). Following the computation chain suggested by Chen et
al., (2018), the output of our encoder is computed as follows:

$\displaystyle\hat{S}_{i}^{lex}$
$\displaystyle=\operatorname{Norm}(S_{i}^{lex})$ (7)
$\displaystyle\hat{h}_{w_{i}}$
$\displaystyle=GRU_{w}(\hat{S}_{i}^{lex},h_{w_{i-1}})$ $\displaystyle
h_{w_{i}}$
$\displaystyle=\operatorname{FFNN}(\operatorname{Norm}(\text{Dropout}(\hat{h}_{w_{i}})+\hat{S}_{i}^{lex}))$

where we use _Norm_ for the layer normalization, and _Dropout_ for a dropout
regularization Srivastava et al., (2014). The other GRU layers of our
architecture, introduced in previous sections, are modified in a similar way.

In this work we don’t use any attention mechanism like in the _Transformer_ ,
context is granted by the use of GRU layers (cf. section 2.1). Our choice is
motivated by the work described in Levy et al., (2018), where it is shown that
RNNs, such like $\operatorname{GRU}$, can encode long contexts as a weighted
mean of the inputs, in a way that can be considered similar to an attention
mechanism. Moreover, using $\operatorname{GRU}$ layers avoid the use of
positional encoding, the sequential structure of an input sentence is encoded
implicitly by the RNNs.222We don’t exclude however that using positional
encoding would improve our architecture

The overall architecture of our model is the same as our previous work
Dinarelli & Grobol, (2019) (cf figure 2), the difference is in the computation
of layer’s output described in previous sections.

$w_{1}$$w_{2}$$w_{3}$encoderencoderencoder$\overleftarrow{decoder}$$\overleftarrow{decoder}$$\overleftarrow{decoder}$$\overrightarrow{decoder}$$\overrightarrow{decoder}$$\overrightarrow{decoder}$$\overrightarrow{e_{1}}$$\overrightarrow{e_{2}}$$\overrightarrow{e_{3}}$$\overleftarrow{e_{1}}$$\overleftarrow{e_{2}}$$\overleftarrow{e_{3}}$
Figure 2: Overall architecture of our model, this is the same as the model
presented in our previous work Dinarelli & Grobol, (2019)

##  3 Evaluation

We evaluate our models on three tasks of sequence labelling:

The French corpus MEDIA Bonneau-Maynard et al., (2006) was created for
evaluating spoken dialog systems in the context of hotel information and
reservation in France. Data have been annotated following a rich concept
ontology. Simple semantic components can be combined to create more precise
and complex semantic units.333For example the component localization (we
translate from French) can be combined with city, relative-distance, general-
relative-localization, street, etc. Concepts span in general more than one
word, the goal is thus to tag words and then reconstruct semantic units from
the tags. Statistics of the corpus MEDIA are shown in figure 1.

This task is modelled as sequence labelling splitting semantic units over
multiple words using the BIO formalism Ramshaw & Marcus, (1995). We already
evaluated on this task in the past, using a large variety of probabilistic
models Dinarelli et al., (2009a, b); Quarteroni et al., (2009); Hahn et al.,
(2010); Dinarelli, (2010); Dinarelli & Rosset, (2011); Dinarelli et al.,
(2011).

In the MEDIA corpus some word classes (clusters) are provided with the
annotation. Such classes allow for a better generalization on words belonging
to categories for which lists can be easily retrieved, or constructed, in the
context of a real human-machine dialog system. Examples of such classes are
the name of cities in France (class _CITY_), the hotel brands (class _HOTEL_),
currency amounts (_AMOUNTS_), etc. We used these classes in some of our
experiments, this is indicated with _FEAT_ in the tables of the following
section.

| Training | Dev | Test
---|---|---|---
# sentences | 12 908 | 1 259 | 3 005
| Words | Concepts | Words | Concepts | Words | Concepts
# tokens | 94 466 | 43 078 | 10 849 | 4 705 | 25 606 | 11 383
dictionary | 2 210 | 99 | 838 | 66 | 1 276 | 78
OOV% | – | – | 1,33 | 0,02 | 1,39 | 0,04
Table 1: Statistics of the French corpus MEDIA

The English Penn TreeBank corpus Marcus et al., (1993), named WSJ henceforth,
is one of the most used corpus for evaluating models for sequence labelling.
The task consists in assigning each word its POS tag. For our experiments we
use the most common data split: sections 0-18 are used as training data,
sections 19-21 as development data, and sections 22-24 are used for testing.
Statistics of this corpus are given in the table 2.

| Training | Dev | Test
---|---|---|---
# sentences | 38 219 | 5 527 | 5 462
| Words | Labels | Words | Labels | Words | Labels
# tokens | 912 344 | – | 131 768 | – | 129 654 | –
dictionary | 43 210 | 45 | 15 081 | 45 | 13 968 | 45
OOV% | – | – | 22,61 | 0 | 20,00 | 0
Table 2: Statistics of the English corpus WSJ

The German corpusTIGER Brants et al., (2004) is annotated with rich morpho-
syntactic information. These include POS tags like in the WSJ corpus, but also
gender, number, cases, and other inflection information, as well as
conjugation information for verbs. The combination of all these pieces of
information constitutes the output labels. This task is similar to the POS
tagging from a sequence labelling point of view, however it is made more
difficult on one side by the language (German has more complex syntactic
structures than English), on the other side by the relatively large number of
labels to be predicted ($694$ in total, while there are $138$ labels in MEDIA
and $45$ in the WSJ corpus). We use the same data split used in Lavergne &
Yvon, (2017). Statistics of this corpus are given in the table 3. As we can
see from tables, the size of dictionaries, for both words and labels, is
relatively large in the WSJ and TIGER corpora.

| Training | Dev | Test
---|---|---|---
# sentences | 40 472 | 5 000 | 5 000
| Words | Labels | Words | Labels | Words | Labels
# tokens | 719 530 | – | 76 704 | – | 92 004 | –
dictionary | 77 220 | 681 | 15 852 | 501 | 20 149 | 537
OOV% | – | – | 30,90 | 0,01 | 37,18 | 0,015
Table 3: Statistics of the German corpus TIGER

We note that the Out-Of-Vocabulary (OOV) rate in the MEDIA corpus is quite
small. Indeed there are only two OOV words in both Dev and Test data, moreover
they are function words. There is basically no unknown word in this task. In
contrast the OOV rate in both the WSJ and the TIGER corpora is quite high:
roughly one word out of five in the first and one out of three in the second.

###  3.1 Settings

We used the corpus MEDIA for fine-tuning the hyper-parameters of our models in
Dinarelli & Grobol, (2019). MEDIA is the smallest corpus among the three used,
and allows thus a faster optimization. The settings used in this work are the
same as those used in Dinarelli et al., (2017); Dinarelli & Grobol, (2019),
where the same data were used for evaluation (except for the TIGER corpus).
The settings for the WSJ corpus are the same, except for the word embeddings
($300$ dimensions) and the learning rate ($2.5\text{\times}{10}^{-4}$). We use
the same settings for WSJ and TIGER. Since the latter corpus is the largest,
we ran few experiments to choose the correct hidden layer size ($300$).

Like we already explained in Dinarelli & Grobol, (2019), at first our
experiments were not fitting in the memory of our GPUs. In order to solve this
problem we found two solutions. In the first we organize the training data as
a single stream of items. This is then split into chunks of fixed size,
shifting each chunk by one token with respect to the previous (thus
consecutive chunks overlap). The second solution is more intuitive and
consists in clustering sentences of the same length. This way, very long
(rare) sentences are assigned to very small clusters, thus fitting in memory,
and small and average clusters form bigger clusters, which still fit in
memory. In our experiments we found out that the first solution works far
better on the MEDIA corpus. On the WSJ and TIGER corpora the two solutions are
equivalent, we thus prefer the second for these two tasks, which is more
intuitive and is expected to work better on bigger corpora.

Our neural network system is coded with
pytorch444https://pytorch.org/docs/stable/index.html, experiments are run
under Linux Debian distribution with GeForce GTX 1080 GPUs.

###  3.2 Results

Model | Accuracy | F1 | CER
---|---|---|---
MEDIA DEV
Seq2Biseq Dinarelli & Grobol, (2019) | 89.97 | 86.57 | 10.42
Seq2Biseq${}_{\text{2-opt}}$ Dinarelli & Grobol, (2019) | 90.22 | 86.88 | 9.97
Seq2Biseq-Transformer${}_{\text{2-opt}}$ | 90.08 | 86.93 | 9.87
Seq2Biseq-Transformer+FEAT${}_{\text{2-opt}}$ | 90.14 | 87.05 | 9.54
MEDIA TEST
BiGRU+CRF Dinarelli et al., (2017) | – | 86.69 | 10.13
LD-RNNdeep Dinarelli et al., (2017) | – | 87.36 | 9.8
Seq2Biseq Dinarelli & Grobol, (2019) | 89.57 | 87.50 | 10.26
Seq2Biseq${}_{\text{2-opt}}$ Dinarelli & Grobol, (2019) | 89.79 | 87.69 | 9.93
Seq2Biseq-Transformer${}_{\text{2-opt}}$ | 89.98 | 87.81 | 9.67
Seq2Biseq-Transformer+FEAT${}_{\text{2-opt}}$ | 90.12 | 87.94 | 9.48
Table 4: Results of different variants of our model for the semantic tagging
task of the corpus MEDIA, compared to the state-of-the-art

Results for the MEDIA task are means over $10$ experiments, the parameters are
initialized at random555Using default initialization of the _Pytorch_ library
for each experiment. On this task we evaluate our models in terms of accuracy
and, since the goal is to re-construct semantic units spanning over multiple
words, also with F1 measure and _Concept Error Rate_ (CER). CER is computed by
aligning the model prediction with the gold annotation with the Levenshtein
distance algorithm, then the sum of insertions, deletions and substitutions is
divided by the length of the gold annotation. On the other two tasks, since
there is no label splitting over multiple words, we evaluate the models only
in terms of accuracy.

Since the model presented in this paper is an evolution of the _Seq2Biseq_
model proposed in Dinarelli & Grobol, (2019), the model proposed here is named
_Seq2Biseq-Transformer_ , as it adds characteristics of the _Transformer_ on
top of the _Seq2Biseq_ model.

Results on Dev and Test data of the MEDIA task are shown in table 4. For
comparison we show also results presented in our previous works Dinarelli et
al., (2017); Dinarelli & Grobol, (2019), which are state-of-the-art on this
task.

As described in Dinarelli & Grobol, (2019), the Seq2Biseq model gives better
results when trained with $2$ optimizers (see Dinarelli & Grobol, (2019) for
details), one for each decoder. The Seq2Biseq-Transformer model is thus always
trained with $2$ optimizers.

As we can see in table 4, results obtained on MEDIA with the Seq2Biseq-
Transformer model are slightly worse than Seq2Biseq’s results in terms of
accuracy, and they are slightly better in terms of F1 and CER.666We did not
perform significance tests Adding word classes available for the MEDIA task
(cf. the begin of section 3, data description) as input to the model, results
improve with all evaluation metrics and constitute the new state-of-the-art on
this task with manual transcription.777SLU is performed on automatic
transcription, in this work however we evaluate our models against sequence
labelling problems.

Model | Accuracy
---|---
| WSJ DEV | WSJ TEST
LD-RNNdeep Dinarelli et al., (2017) | 96.90 | 96.91
LSTM+CRF Ma & Hovy, (2016) | – | 97.13
Seq2Biseq Dinarelli & Grobol, (2019) | 97.13 | 97.20
Seq2Biseq2-opt Dinarelli & Grobol, (2019) | 97.33 | 97.35
Seq2Biseq-Transformer2-opt | 97.33 | 97.36
Models using pre-trained embeddings888In Guo et al., (2019) it is actually not
clear if the Transformer and Star-Transformer models have been trained with or
without pre-trained embeddings. Authors report “ _Advanced Techniques_ except
widely-used pre-trained embeddings” in the caption of table 5. However results
are compared with literature models using pre-trained embeddings, and this is
not mentioned in the text.
Transformer+Char Guo et al., (2019) | – | 97.14
LSTM+CRF + Glove Ma & Hovy, (2016) | 97.46 | 97.55
LSTM+LD-RNN + Glove Zhang et al., (2018b) | – | 97.59
Star-Transformer+Char Guo et al., (2019) | – | 97.64
Table 5: Results of different variants of our model for the POS tagging task
on the WSJ corpus, compared to the state-of-the-art

Results of POS tagging on the WSJ corpus are presented in table 5. As we can
see in this case, adding functionalities of the _Transformer_ to the Seq2Biseq
model give exactly the same result as the Seq2Biseq model on the Dev data, and
slightly better on the Test data (on a single experiment).

Keeping both these results and those on the MEDIA task into account, we could
conclude that the Seq2Biseq-Transformer model does not really improve our
previous model. However we would like to point out that the main goal of this
work was to integrate characteristics of the _Transformer_ model, and to come
up with a generic architecture, which can be applied not only to traditional
sequence labelling tasks, but also to tasks such like machine translation and
syntactic parsing. Indeed the Seq2Biseq-Transformer architecture has been
modified, with respect to our previous model, in order to perform all these
tasks. We consider that obtaining similar results (which are still state-of-
the-art) with a more versatile model is still a positive result. Additionally,
observing the training procedure of the Seq2Biseq-Transformer model, we found
out that while final results are similar, the loss values on both training and
Dev data are better than those obtained with the Seq2Biseq model. This could
actually mean the model is over-fitting, and we speculate that instead of
using the same settings as in Dinarelli & Grobol, (2019), since the Seq2Biseq-
Transformer is more complex (at least in terms of parameters, and see below),
a fine-tuning of parameters could lead to better performances.

In table 5 we can see that, whether we use one or two decoders for training,
our models improve the _LSTM+CRF_ model Ma & Hovy, (2016) without using the
embeddings pre-trained with _GloVe_ Pennington et al., (2014). For a matter of
comparison, we show also the best results on this task, obtained with pre-
trained embeddings. Though our results are lower than state-of-the-art in
terms of accuracy, they are not too far, proving that our hybrid neural
architecture is quite effective. In particular, we compare our results with
those reported in Guo et al., (2019) obtained with the _Transformer_ and
_Star-Transformer_ models.999In Guo et al., (2019) it is actually not clear if
the Transformer and Star-Transformer models have been trained with or without
pre-trained embeddings. Authors report “ _Advanced Techniques_ except widely-
used pre-trained embeddings” in the caption of table 5. However results are
compared with literature models using pre-trained embeddings, and this is not
mentioned in the text. Whether the _Transformer_ model has been trained with
or without pre-trained embeddings, our models outperform it by a significant
margin. As we mentioned in section 2.5, we have experienced a drop of 3 points
in the F1 measure using a _Transformer_ for mention detection in preliminary
experiments (not reported in this work). We hypothesize that the one-to-one
correspondence between input and output units in sequence labelling tasks is a
very useful information, and the lack of performance of the _Transformer_ on
sequence labelling tasks can be explained by the fact that this model doesn’t
use it.

Model | Accuracy
---|---
| TIGER DEV | TIGER TEST
Seq2Biseq2-opt |  93.90 (98.30) |  91.86 (97.74)
VO-CRF Lavergne & Yvon, (2017) | – | 88.78
Table 6: Results of different variants of our neural architecture for the
morpho-syntactic tagging task on the TIGER corpus, compared with state-of-the-
art

In table 6 we show results obtained on the morpho-syntactic tagging task on
the German corpus TIGER. To the best of our knowledge, the best results in the
literature for this task are reported in Lavergne & Yvon, (2017), obtained
using a _Variable Order_ CRF (VO-CRF). We can see thus that our model improves
the state-of-the-art on this task. It is important to note however that the
TIGER corpus has been less used for the evaluation of neural models for
sequence labelling. Results reported in Lavergne & Yvon, (2017) are indeed
obtained with a discrete-representation based, though quite sophisticated, CRF
model. In table 6 we report also, between parenthesis, POS tagging results.
These are obtained from morpho-syntactic tagging results extracting the POS
tag from the complete label. That is no model training has been performed for
the POS tagging task alone.

##  4 Conclusions

In this paper we proposed a neural model for sequence labelling integrating
characteristics of the most successful models of the last couple of years:
bidirectional RNNs, _Sequence-to-Sequence_ model, and the _Transfomer_ model.
An evaluation on three tasks of sequence labelling shows that the proposed
model is very effective for this kind of tasks. Indeed it achieves results
comparable to the state-of-the-art, but keeping a generic architecture that
allows to use the same model on other kind of tasks like machine translation
and syntactic parsing. Moreover in some cases our model outperforms state-of-
the-art models, proving that our hybrid architecture has its place among
current neural architectures.

## Acknowledgments

This research is part of the program “Investissements d’Avenir” funded by the
French National Research Agency (ANR) ANR-10-LABX-0083 (Labex EFL).

This work has been partially funded by the French ANR project DEMOCRAT
(_“Description et modélisation des chaînes de référence : outils pour
l’annotation de corpus et le traitement automatique”_) project
ANR-15-CE38-0008.

## References

  * Bahdanau et al., (2014) Bahdanau D., Cho K. & Bengio Y. (2014).  Neural machine translation by jointly learning to align and translate.  CoRR, abs/1409.0473.
  * Bengio et al., (2003) Bengio Y., Ducharme R., Vincent P. & Jauvin C. (2003).  A neural probabilistic language model.  JOURNAL OF MACHINE LEARNING RESEARCH, 3, 1137–1155.
  * Bonneau-Maynard et al., (2006) Bonneau-Maynard H., Ayache C., Bechet F., Denis A., Kuhn A., Lefèvre F., Mostefa D., Qugnard M., Rosset S. & Servan, S. Vilaneau J. (2006).  Results of the french evalda-media evaluation campaign for literal understanding.  In LREC, p. 2054–2059, Genoa, Italy.
  * Brants et al., (2004) Brants S., Dipper S., Eisenberg P., Hansen-Schirra S., Konig E., Lezius W., Rohrer C., Smith G. & Uszkoreit H. (2004).  TIGER: Linguistic interpretation of a german corpus.  Research on Language and Computation, 2(4), 597–620.
  * Chen et al., (2018) Chen M. X., Firat O., Bapna A., Johnson M., Macherey W., Foster G., Jones L., Schuster M., Shazeer N., Parmar N., Vaswani A., Uszkoreit J., Kaiser L., Chen Z., Wu Y. & Hughes M. (2018).  The Best of Both Worlds: Combining Recent Advances in Neural Machine Translation.  In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, volume 1, p. 76–86: Association for Computational Linguistics.
  * Cho et al., (2014) Cho K., van Merrienboer B., Gulcehre C., Bahdanau D., Bougares F., Schwenk H. & Bengio Y. (2014).  Learning Phrase Representations using RNN Encoder–Decoder for Statistical Machine Translation.  In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), p. 1724–1734: Association for Computational Linguistics.
  * Collins, (1997) Collins M. (1997).  Three generative, lexicalised models for statistical parsing.  In Proceedings of ACL, p. 16–23, Stroudsburg, PA, USA: Association for Computational Linguistics.
  * Collobert et al., (2011) Collobert R., Weston J., Bottou L., Karlen M., Kavukcuoglu K. & Kuksa P. (2011).  Natural language processing (almost) from scratch.  J. Mach. Learn. Res., 12.
  * Dai et al., (2019) Dai Z., Yang Z., Yang Y., Carbonell J., Le Q. V. & Salakhutdinov R. (2019).  Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context.  arXiv preprint 1901.02860.
  * De Mori et al., (2008) De Mori R., Bechet F., Hakkani-Tur D., McTear M., Riccardi G. & Tur G. (2008).  Spoken language understanding: A survey.  IEEE Signal Processing Magazine, 25, 50–58.
  * Dehghani et al., (2018) Dehghani M., Gouws S., Vinyals O., Uszkoreit J. & Kaiser L. (2018).  Universal transformers.  CoRR, abs/1807.03819.
  * Devlin et al., (2018) Devlin J., Chang M.-W., Lee K. & Toutanova K. (2018).  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding.  arXiv preprint 1810.04805.
  * Dinarelli, (2010) Dinarelli M. (2010).  Spoken Language Understanding: from Spoken Utterances to Semantic Structures.  PhD thesis, International Doctoral School in Information and Communication Technology, Dipartimento di Ingegneria e Scienza dell’ Informazione, via Sommarive 14, 38100 Povo di Trento (TN), Italy.
  * Dinarelli & Grobol, (2019) Dinarelli M. & Grobol L. (2019).  Seq2biseq: Bidirectional output-wise recurrent neural networks for sequence modelling.  CoRR, abs/1904.04733.
  * Dinarelli et al., (2009a) Dinarelli M., Moschitti A. & Riccardi G. (2009a).  Concept segmentation and labeling for conversational speech.  In Proceedings of the International Conference of the Speech Communication Assosiation (Interspeech), Brighton, U.K.
  * Dinarelli et al., (2009b) Dinarelli M., Moschitti A. & Riccardi G. (2009b).  Re-ranking models based on small training data for spoken language understanding.  In Conference of Empirical Methods for Natural Language Processing, p. 11–18, Singapore.
  * Dinarelli et al., (2011) Dinarelli M., Moschitti A. & Riccardi G. (2011).  Discriminative reranking for spoken language understanding.  IEEE TASLP, 20, 526–539.
  * Dinarelli & Rosset, (2011) Dinarelli M. & Rosset S. (2011).  Hypotheses selection criteria in a reranking framework for spoken language understanding.  In Conference of Empirical Methods for Natural Language Processing, p. 1104–1115, Edinburgh, U.K.
  * Dinarelli & Rosset, (2012a) Dinarelli M. & Rosset S. (2012a).  Tree representations in probabilistic models for extended named entity detection.  In European Chapter of the Association for Computational Linguistics (EACL), p. 174–184, Avignon, France.
  * Dinarelli & Rosset, (2012b) Dinarelli M. & Rosset S. (2012b).  Tree-structured named entity recognition on ocr data: Analysis, processing and results.  In N. C. C. Chair), K. Choukri, T. Declerck, M. U. Dogan, B. Maegaard, J. Mariani, J. Odijk & S. Piperidis, Eds., Proceedings of the Eight International Conference on Language Resources and Evaluation (LREC’12), Istanbul, Turkey: European Language Resources Association (ELRA).
  * Dinarelli & Tellier, (2016a) Dinarelli M. & Tellier I. (2016a).  Improving recurrent neural networks for sequence labelling.  CoRR, abs/1606.02555.
  * Dinarelli & Tellier, (2016b) Dinarelli M. & Tellier I. (2016b).  New recurrent neural network variants for sequence labeling.  In Proceedings of the 17th International Conference on Intelligent Text Processing and Computational Linguistics, Konya, Turkey: Lecture Notes in Computer Science (Springer).
  * Dinarelli et al., (2017) Dinarelli M., Vukotic V. & Raymond C. (2017).  Label-dependency coding in Simple Recurrent Networks for Spoken Language Understanding.  In Interspeech, Stockholm, Sweden.
  * Dupont et al., (2017) Dupont Y., Dinarelli M. & Tellier I. (2017).  Label-dependencies aware recurrent neural networks.  In Proceedings of CICling, Budapest, Hungary: LNCS, Springer.
  * Grouin et al., (2011) Grouin C., Dinarelli M., Rosset S., Wisniewski G. & Zweigenbaum P. (2011).  Coreference resolution in clinical reports. the limsi participation in the i2b2/va 2011 challenge.  In In Proceedings of i2b2/VA 2011 Coreference Resolution Workshop.
  * Guo et al., (2019) Guo Q., Qiu X., Liu P., Shao Y., Xue X. & Zhang Z. (2019).  Star-transformer.  CoRR, abs/1902.09113.
  * Hahn et al., (2010) Hahn S., Dinarelli M., Raymond C., Lefèvre F., Lehen P., De Mori R., Moschitti A., Ney H. & Riccardi G. (2010).  Comparing stochastic approaches to spoken language understanding in multiple languages.  IEEE TASLP, 99.
  * Lample et al., (2016) Lample G., Ballesteros M., Subramanian S., Kawakami K. & Dyer C. (2016).  Neural Architectures for Named Entity Recognition.  In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, p. 260–270: Association for Computational Linguistics.
  * Lavergne & Yvon, (2017) Lavergne T. & Yvon F. (2017).  Learning the structure of variable-order crfs: a finite-state perspective.  In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, p. 433–439: Association for Computational Linguistics.
  * Levy et al., (2018) Levy O., Lee K., FitzGerald N. & Zettlemoyer L. (2018).  Long short-term memory as a dynamically computed element-wise weighted sum.  In Proceedings of ACL, p. 732–739: ACL.
  * Ma & Hovy, (2016) Ma X. & Hovy E. (2016).  End-to-end sequence labeling via bi-directional lstm-cnns-crf.  In Proceedings of ACL.
  * Marcus et al., (1993) Marcus M. P., Santorini B. & Marcinkiewicz M. A. (1993).  Building a large annotated corpus of english: The penn treebank.  COMPUTATIONAL LINGUISTICS, 19(2).
  * Ng & Cardie, (2002) Ng V. & Cardie C. (2002).  Improving Machine Learning Approcahes to Corefrence Resolution.  In Proceedings of ACL’02, p. 104–111.
  * Pennington et al., (2014) Pennington J., Socher R. & Manning C. D. (2014).  Glove: Global vectors for word representation.  In Empirical Methods in Natural Language Processing (EMNLP), p. 1532–1543.
  * Peters et al., (2018) Peters M., Neumann M., Iyyer M., Gardner M., Clark C., Lee K. & Zettlemoyer L. (2018).  Deep Contextualized Word Representations.  In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, volume 1, p. 2227–2237: Association for Computational Linguistics.
  * Quarteroni et al., (2009) Quarteroni S., Riccardi G. & Dinarelli M. (2009).  What’s in an ontology for spoken language understanding.  In Proceedings of the International Conference of the Speech Communication Assosiation (Interspeech), Brighton, U.K.
  * Ramshaw & Marcus, (1995) Ramshaw L. & Marcus M. (1995).  Text chunking using transformation-based learning.  In Proceedings of the 3rd Workshop on Very Large Corpora, p. 84–94, Cambridge, MA, USA.
  * Rush et al., (2012) Rush A. M., Reichart R., Collins M. & Globerson A. (2012).  Improved parsing and pos tagging using inter-sentence consistency constraints.  In Proceedings of EMNLP-CoNLL, Stroudsburg, PA, USA.
  * Soon et al., (2001) Soon W. M., Ng H. T. & Lim D. C. Y. (2001).  A Machine Learning Approach to Coreference Resolution of Noun Phrases.  Computational Linguistics, 27(4), 521–544.
  * Srivastava et al., (2014) Srivastava N., Hinton G., Krizhevsky A., Sutskever I. & Salakhutdinov R. (2014).  Dropout: A simple way to prevent neural networks from overfitting.  Journal of Machine Learning Research, 15, 1929–1958.
  * Sutskever et al., (2014) Sutskever I., Vinyals O. & Le Q. V. (2014).  Sequence to sequence learning with neural networks.  In Proceedings of NIPS, Cambridge, MA, USA: MIT Press.
  * Vaswani et al., (2017) Vaswani A., Shazeer N., Parmar N., Uszkoreit J., Jones L., Gomez A. N., Kaiser L. & Polosukhin I. (2017).  Attention is All you Need.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan & R. Garnett, Eds., Advances in Neural Information Processing Systems 30, p. 5998–6008. Curran Associates, Inc.
  * Vinyals et al., (2015) Vinyals O., Kaiser L., Koo T., Petrov S., Sutskever I. & Hinton G. (2015).  Grammar As a Foreign Language.  In Proceedings of the 28th International Conference on Neural Information Processing, volume 2 of NIPS’15, p. 2773–2781, Cambridge, MA, USA: MIT Press.
  * Xia et al., (2017) Xia Y., Tian F., Wu L., Lin J., Qin T., Yu N. & Liu T.-Y. (2017).  Deliberation networks: Sequence generation beyond one-pass decoding.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan & R. Garnett, Eds., Advances in Neural Information Processing Systems 30, p. 1784–1794. Curran Associates, Inc.
  * Zhang et al., (2018a) Zhang X., Su J., Qin Y., Liu Y., Ji R. & Wang H. (2018a).  Asynchronous bidirectional decoding for neural machine translation.  CoRR, abs/1801.05122.
  * Zhang et al., (2018b) Zhang Y., Chen H., Zhao Y., Liu Q. & Yin D. (2018b).  Learning tag dependencies for sequence tagging.  In International Joint Conference on Artificial Intelligence (IJCAI).


††thanks: Deceased 30 March 2017††thanks: Ref. author e-mail:
<EMAIL_ADDRESS>Deceased 30 September 2012††thanks: Ref.
author e-mail<EMAIL_ADDRESS>
# LISA Pathfinder Performance Confirmed in an Open-Loop Configuration: Results
from the Free-Fall Actuation Mode

M. Armano  European Space Technology Centre, European Space Agency, Keplerlaan
1, 2200 AG Noordwijk, The Netherlands  H. Audley  Albert-Einstein-Institut,
Max-Planck-Institut für Gravitationsphysik und Leibniz Universität Hannover,
Callinstraße 38, 30167 Hannover, Germany  J. Baird  APC, Univ Paris Diderot,
CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris Cité, France  P. Binetruy
APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris
Cité, France  M. Born  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  D. Bortoluzzi  Department of Industrial Engineering,
University of Trento, via Sommarive 9, 38123 Trento, and Trento Institute for
Fundamental Physics and Application / INFN  E. Castelli  Dipartimento di
Fisica, Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  A. Cavalleri  Istituto di
Fotonica e Nanotecnologie, CNR-Fondazione Bruno Kessler, I-38123 Povo, Trento,
Italy  A. Cesarini  DISPEA, Università di Urbino “Carlo Bo”, Via S. Chiara, 27
61029 Urbino/INFN, Italy  A. M. Cruise  The School of Physics and Astronomy,
University of Birmingham, Birmingham, UK  K. Danzmann  Albert-Einstein-
Institut, Max-Planck-Institut für Gravitationsphysik und Leibniz Universität
Hannover, Callinstraße 38, 30167 Hannover, Germany  M. de Deus Silva  European
Space Astronomy Centre, European Space Agency, Villanueva de la Cañada, 28692
Madrid, Spain  I. Diepholz  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  G. Dixon  The School of Physics and Astronomy, University
of Birmingham, Birmingham, UK  R. Dolesi  Dipartimento di Fisica, Università
di Trento and Trento Institute for Fundamental Physics and Application / INFN,
38123 Povo, Trento, Italy  L. Ferraioli  Institut für Geophysik, ETH Zürich,
Sonneggstrasse 5, CH-8092, Zürich, Switzerland  V. Ferroni  Dipartimento di
Fisica, Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  E. D. Fitzsimons  The UK
Astronomy Technology Centre, Royal Observatory, Edinburgh, Blackford Hill,
Edinburgh, EH9 3HJ, UK  M. Freschi  European Space Astronomy Centre, European
Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain  L. Gesa  Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n,
08193 Cerdanyola del Vallès, Spain  Institut d’Estudis Espacials de Catalunya
(IEEC), C/ Gran Capità 2-4, 08034 Barcelona, Spain  F. Gibert  Dipartimento di
Fisica, Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  isardSAT SL, Marie Curie 8-14,
08042 Barcelona, Catalonia, Spain  D. Giardini  Institut für Geophysik, ETH
Zürich, Sonneggstrasse 5, CH-8092, Zürich, Switzerland  R. Giusteri  Albert-
Einstein-Institut, Max-Planck-Institut für Gravitationsphysik und Leibniz
Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  Dipartimento
di Fisica, Università di Trento and Trento Institute for Fundamental Physics
and Application / INFN, 38123 Povo, Trento, Italy  C. Grimani  DISPEA,
Università di Urbino “Carlo Bo”, Via S. Chiara, 27 61029 Urbino/INFN, Italy
J. Grzymisch  European Space Technology Centre, European Space Agency,
Keplerlaan 1, 2200 AG Noordwijk, The Netherlands  I. Harrison  European Space
Operations Centre, European Space Agency, 64293 Darmstadt, Germany  M-S.
Hartig  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  G.
Heinzel  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  M.
Hewitson  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  D.
Hollington  High Energy Physics Group, Physics Department, Imperial College
London, Blackett Laboratory, Prince Consort Road, London, SW7 2BW, UK  D.
Hoyland  The School of Physics and Astronomy, University of Birmingham,
Birmingham, UK  M. Hueller  Dipartimento di Fisica, Università di Trento and
Trento Institute for Fundamental Physics and Application / INFN, 38123 Povo,
Trento, Italy  H. Inchauspé  APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu,
Obs de Paris, Sorbonne Paris Cité, France  Department of Mechanical and
Aerospace Engineering, MAE-A, P.O. Box 116250, University of Florida,
Gainesville, Florida 32611, USA  O. Jennrich  European Space Technology
Centre, European Space Agency, Keplerlaan 1, 2200 AG Noordwijk, The
Netherlands  P. Jetzer  Physik Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  N. Karnesis  APC, Univ
Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris Cité, France
B. Kaune  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  N.
Korsakova  SUPA, Institute for Gravitational Research, School of Physics and
Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK  C. J. Killow  SUPA,
Institute for Gravitational Research, School of Physics and Astronomy,
University of Glasgow, Glasgow, G12 8QQ, UK  J. A. Lobo  Institut de Ciències
de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193
Cerdanyola del Vallès, Spain  Institut d’Estudis Espacials de Catalunya
(IEEC), C/ Gran Capità 2-4, 08034 Barcelona, Spain  L. Liu  Dipartimento di
Fisica, Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  J. P. López-Zaragoza  Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n,
08193 Cerdanyola del Vallès, Spain  R. Maarschalkerweerd  European Space
Operations Centre, European Space Agency, 64293 Darmstadt, Germany  D. Mance
Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, CH-8092, Zürich,
Switzerland  N. Meshksar  Institut für Geophysik, ETH Zürich, Sonneggstrasse
5, CH-8092, Zürich, Switzerland  V. Martín  Institut de Ciències de l’Espai
(ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del
Vallès, Spain  Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran
Capità 2-4, 08034 Barcelona, Spain  L. Martin-Polo  European Space Astronomy
Centre, European Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain
J. Martino  APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris,
Sorbonne Paris Cité, France  F. Martin-Porqueras  European Space Astronomy
Centre, European Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain
I. Mateos  Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de
Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain  Institut d’Estudis
Espacials de Catalunya (IEEC), C/ Gran Capità 2-4, 08034 Barcelona, Spain  P.
W. McNamara  European Space Technology Centre, European Space Agency,
Keplerlaan 1, 2200 AG Noordwijk, The Netherlands  J. Mendes  European Space
Operations Centre, European Space Agency, 64293 Darmstadt, Germany  L. Mendes
European Space Astronomy Centre, European Space Agency, Villanueva de la
Cañada, 28692 Madrid, Spain  M. Nofrarias  Institut de Ciències de l’Espai
(ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del
Vallès, Spain  Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran
Capità 2-4, 08034 Barcelona, Spain  S. Paczkowski  Albert-Einstein-Institut,
Max-Planck-Institut für Gravitationsphysik und Leibniz Universität Hannover,
Callinstraße 38, 30167 Hannover, Germany  M. Perreur-Lloyd  SUPA, Institute
for Gravitational Research, School of Physics and Astronomy, University of
Glasgow, Glasgow, G12 8QQ, UK  A. Petiteau  APC, Univ Paris Diderot,
CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris Cité, France  P. Pivato
Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  E.
Plagnol  APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne
Paris Cité, France  J. Ramos-Castro  Department d’Enginyeria Electrònica,
Universitat Politècnica de Catalunya, 08034 Barcelona, Spain  Institut
d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capità 2-4, 08034 Barcelona,
Spain  J. Reiche  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  D. I. Robertson  SUPA, Institute for Gravitational
Research, School of Physics and Astronomy, University of Glasgow, Glasgow, G12
8QQ, UK  F. Rivas  Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB,
Carrer de Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain  Institut
d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capità 2-4, 08034 Barcelona,
Spain  G. Russano  Dipartimento di Fisica, Università di Trento and Trento
Institute for Fundamental Physics and Application / INFN, 38123 Povo, Trento,
Italy  J. Slutsky  Gravitational Astrophysics Lab, NASA Goddard Space Flight
Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA  C. F. Sopuerta  Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n,
08193 Cerdanyola del Vallès, Spain  Institut d’Estudis Espacials de Catalunya
(IEEC), C/ Gran Capità 2-4, 08034 Barcelona, Spain  T. Sumner  High Energy
Physics Group, Physics Department, Imperial College London, Blackett
Laboratory, Prince Consort Road, London, SW7 2BW, UK  D. Texier  European
Space Astronomy Centre, European Space Agency, Villanueva de la Cañada, 28692
Madrid, Spain  J. I. Thorpe  Gravitational Astrophysics Lab, NASA Goddard
Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA  D. Vetrugno
Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  S.
Vitale  Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  G.
Wanner  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  H.
Ward  SUPA, Institute for Gravitational Research, School of Physics and
Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK  P. J. Wass  High
Energy Physics Group, Physics Department, Imperial College London, Blackett
Laboratory, Prince Consort Road, London, SW7 2BW, UK  Department of Mechanical
and Aerospace Engineering, MAE-A, P.O. Box 116250, University of Florida,
Gainesville, Florida 32611, USA  W. J. Weber  Dipartimento di Fisica,
Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  L. Wissel  Albert-Einstein-
Institut, Max-Planck-Institut für Gravitationsphysik und Leibniz Universität
Hannover, Callinstraße 38, 30167 Hannover, Germany  A. Wittchen  Albert-
Einstein-Institut, Max-Planck-Institut für Gravitationsphysik und Leibniz
Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  P. Zweifel
Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, CH-8092, Zürich,
Switzerland

(August 28, 2024)

###### Abstract

We report on the results of the LISA Pathfinder (LPF) free-fall mode
experiment, in which the control force needed to compensate the quasistatic
differential force acting on two test masses is applied intermittently as a
series of “impulse” forces lasting a few seconds and separated by roughly 350
s periods of true free fall. This represents an alternative to the normal LPF
mode of operation in which this balancing force is applied continuously, with
the advantage that the acceleration noise during free fall is measured in the
absence of the actuation force, thus eliminating associated noise and force
calibration errors. The differential acceleration noise measurement presented
here with the free-fall mode agrees with noise measured with the continuous
actuation scheme, representing an important and independent confirmation of
the LPF result. An additional measurement with larger actuation forces also
shows that the technique can be used to eliminate actuation noise when this is
a dominant factor.

_Introduction.—_ LISA Pathfinder (LPF) McNamara  _et al._ (2008) was a
differential accelerometer designed to demonstrate the free fall of geodesic
reference test masses (TMs) at the level required for space-borne
gravitational wave observatories such as LISA Amaro-Seoane  _et al._ . LPF
achieved this by using a high precision interferometer to measure the relative
acceleration, $\Delta g$, between two TMs placed in the same spacecraft (SC),
along the $x$ axis joining their centers (see Fig. 1). LISA is a truly open-
loop differential acceleration measurement, with both TMs unforced inside
separate drag-free spacecrafts. In LPF closed-loop forces must be employed to
keep the two TMs inside a single spacecraft, and this applied force is part of
the measurement. Indeed, it is not possible for both TMs to be in free fall
along $x$ at the same time, like would be in LISA.

In the normal LPF operations conditions, the observable $\Delta g$ is measured
by applying a calibrated compensation force $g_{c}$ on TM2 (all forces are
expressed here per unit mass) and is extracted according to $\Delta
g\simeq\Delta\ddot{x}-g_{c}$, with $\Delta\ddot{x}$ being the numerical second
time derivative of the relative displacement $\Delta x$. The compensation is
exerted by an electrostatic control loop continuously acting on TM2 with unity
gain around 1 mHz. The reconstructed signal for $\Delta g$ is dominated by
$g_{c}$ for frequencies roughly below the 1 mHz band of the controller, while
$\Delta\ddot{x}$ leads at higher frequencies where the TM is essentially free.
The resulting time series for $\Delta g$ depends on the actuator calibration
Armano  _et al._ (2016a). In addition, the voltage noise of the actuator that
applies $g_{c}$ introduces an extra force noise that was expected to be
dominant at low frequencies Antonucci  _et al._ (2011).

To measure acceleration noise in a LISA-like configuration without $x$ axis
applied forces, a dedicated noise measurement using intermittent free fall has
been designed Grynagier  _et al._ (2009). This alternative technique aims at
estimating the residual noise in $\Delta g$ independent of the actuator
calibration and free of actuation noise, and to characterize, by comparison,
the contribution of actuation noise measured in standard operations. This
configuration was tested in the LPF free-fall mode (or “drift” mode)
experiment in which the compensation force on TM2 is applied intermittently in
the form of high amplitude pulses with period of a few seconds, in between
which the TM is let to fly with no compensation force along $x$.

The free-fall mode provides a measurement of the noise in $\Delta g$ that
coincides with that measured in the standard LPF configuration with continuous
control and thus it confirms, as an independent measurement, the LPF
performance. Indeed, actuation noise measured in flight conditions was a not a
dominant contributor around 1 mHz, thus removing the $x$ actuator produced a
small effect on the acceleration noise spectrum. Moreover, the presented
result demonstrates the functionality of an alternative control for space-
based gradiometers Rummel  _et al._ (2011), where force gradients are measured
from the applied compensation force needed to hold the TM steady.

Figure 1: LPF capacitive actuation along $x$ and housing coordinate systems.
$g_{1}$ and $g_{2}$ indicate the stray acceleration experienced by TM1 and
TM2, respectively.

_LISA Pathfinder instrument.—_ Two gold-platinum cubic test-masses separated
by $\sim 38$ cm form the core instrument of LPF Carbone  _et al._ (2005). Both
are in free fall inside a single SC with no mechanical contact and each of
them is contained within an electrode housing Dolesi  _et al._ (2003), which
serves as a 6 degree-of-freedom capacitive sensor and electrostatic force and
torque actuator. TM2 is forced by an electrostatic suspension control loop to
stay at a fixed distance from TM1, along $x$ and thus centered in its own
electrode housing. A second controller, called drag free, feeds the thrusters
to keep the SC to follow TM1.

Given the quadratic dependence of force on voltage, the force fluctuation
associated with an actuation voltage amplitude fluctuation depends on the
force levels applied by each electrode. The same four electrodes actuate in
$x$ and $\phi$ (see Fig. 1), with an actuation scheme that keeps the stiffness
constant Antonucci  _et al._ (2011) according to the maximum net forces and
torques allowed, called “authorities”. The resulting $x$-force noise from
actuation amplitude fluctuations depends both on the net applied $x$/$\phi$
force and on the $x$/$\phi$ authorities (the actuation scheme and noise model
are presented in an upcoming publication).

Based on a preflight analysis that considered $\Delta g_{DC}=650$ pm/s2 —based
on conservative gravitational balance precision estimates Armano  _et al._
(2016b) and measured actuation amplitudes between 3 and 8 ppm/Hz1/2 —actuation
amplitude fluctuations were considered as the leading low frequency
acceleration noise source for LPF at roughly 7 fm/s2/Hz1/2 at 1 mHz Antonucci
_et al._ (2011) (this analysis considered $\Delta g_{DC}$ $\approx 650$ nm/s2
and $\phi$ DC angular accelerations of 2 nrad/s2, with 10$\%$ larger actuation
authorities to accommodate transient dynamics). Over the mission, different
levels of force and torque authority were implemented, beginning with the
_Nominal_ configuration programmed before flight to accommodate potentially
large gravitational imbalances, with $x$-force authority of 1100 pm/s2 (see
Table 1). The in-flight observed DC force imbalance was much smaller, always
below 20 pm/s2 along $x$ Armano  _et al._ (2016a) with angular accelerations
of -1.1 and 0.2 nrad/s2 for TM1 and TM2. This allowed reducing the authorities
from _Nominal_ to the URLA configuration levels, with 26 pm/s2 $x$-force
authority (see Table 1). In this configuration, used for the measurements that
established the published LPF differential acceleration noise floor Armano
_et al._ (2016a, 2018a), the actuation noise, as estimated from a dedicated
in-flight measurement campaign employing various force levels, is less than
20$\%$ of the total acceleration noise power measured over the 0.1 to 1mHz
band Weber  _et al._ (see the dashed line in Fig. 4).

Removing the $x$-axis actuation with the free-fall mode thus, in these flight
conditions, is expected to have only a small impact on the measured
acceleration noise (we note that during the free-fall mode the $\phi$
actuation torque is still applied continuously). Nevertheless, the free-fall
mode experiment still represents an independent measurement of the
differential acceleration without any actuator, immune to possible actuation
nonlinearities or calibration inaccuracies.

_Experiment description and calibration.—_ The free-fall mode implemented on
LPF is a special actuation scheme where the electrostatic control on TM2 is
switched on the sensitive $x$ axis only for a very short duration ($\leq$ 5
s). In particular, an impulse controller tracks the TM2 displacement, $x_{2}$,
during the flight and estimates the impulse necessary to push it back on the
other side against the static field it experiences on board the SC. Then, the
impulse-flight cycle is repeated (see Fig. 2) Grynagier  _et al._ (2009);
Giusteri  _et al._ (2017). The flight interval, $T_{\rm flight}$, is set by
the maximum displacement allowed along $x_{2}$ ($\approx 10\,\mu$m), based on
the preflight estimate of the gravitational imbalance. The experiments
presented here are implemented with a fixed experimental time,
$T_{\rm{exp}}=T_{\rm{flight}}+T_{\rm{imp}}$, of 350.2 s, while impulse
durations ($T_{\rm{imp}}$) of 1 s and 5 s were used in the two measurements.
Figure (2) depicts the start of the first experiment with free-fall mode
performed with $T_{\rm imp}=1$ s and following a noise run executed in
continuous control mode. As visible in the middle panel, the free-fall mode is
characterized by a wide dynamic range in displacement (tens of nm), in
contrast with the continuous mode (tens of pm Giusteri  _et al._ (2017)).

Figure 2: Time series of TM2 force (top), relative displacement, $\Delta x$,
(middle) and $\Delta g$ (bottom) measured in June 2016. The thin lines refer
to a noise measurement with continuous control on TM2 in URLA authority, while
the thick lines indicate a free-fall mode measurement in _nominal_ authority.
The discontinuity in the top panel stems from the use of different telemetry
packets. The transient phase between the two runs is discarded in the bottom
panel.

The main observable of LPF, $\Delta g$, is calculated with free-fall data as
follows:

$\displaystyle\Delta g(t)$
$\displaystyle\equiv\Delta\ddot{x}(t)+\omega_{2}^{2}\Delta x(t)-g_{\rm
rot}(t)$ (1)

where $\omega_{2}^{2}$ is the electrostatic force gradient (“stiffness”),
coupling TM2 to the SC and $g_{\rm rot}(t)$ is the contribution of the
inertial forces acting on the TMs which are described and calculated in Ref.
Armano  _et al._ (2018b). Differently from the definition of $\Delta g$ in
Ref. Armano  _et al._ (2018a), the control force on TM2 is excluded in Eq.
(1), being zero by definition in free-fall mode. In addition, the differential
stiffness coupling the SC motion to $\Delta g$ is neglected in our analysis,
as it is too small to impact the result.

To retrieve the stiffness on TM2, $\omega_{2}^{2}$, we fit $\Delta\ddot{x}$ to
$-\omega_{2}^{2}\Delta x$ flight by flight, as described in Ref. Giusteri  _et
al._ (2017). The resulting parameter values are averaged over the flights to
get a single estimate. Then, the inertial contribution is subtracted from the
residuals of the fit [see Eq. (1)] according to the procedure explained in
Ref. Armano  _et al._ (2018b).

_Measurement data set.—_ The free-fall mode experiment was performed seven
times between June and December 2016, with stable and reliable control
operation in various actuation configurations. This Letter presents the one-
day measurement executed in June with $\phi$ authority based on preflight
analysis (_nominal_ authority) and the last run, with one week duration,
performed in December with lower authority levels on $\phi$ (URLA authority).
The intermediate measurements were used for planning the last long run, which
was implemented to limit the flight amplitude within tens of nm. Indeed, the
large dynamic range achieved in free-fall mode, compared to the continuous
control mode, impacts the interferometer readout. In addition, it increases
timing error issues, as observed also in the dedicated on-ground testing
campaign performed with a torsion pendulum facility Russano  _et al._ (2018a).
In this context, to reduce the gravitational imbalance between the TMs
measured in December, TM1 was actuated along $x$ with a constant out of loop
force with amplitude of 11.2 pN, which was then subtracted from $\Delta g$. It
has been verified that this force does not introduce significant noise or
calibration errors.

Table 1 reports details and calibration results of the two free-fall mode
measurements presented here.

Authority scheme |  $g_{x_{2}}^{\rm max}$ |  $g_{\phi_{1}}^{\rm max}$ |  $g_{\phi_{2}}^{\rm max}$ |  Start date |  $\Delta t$ |  $\Delta x_{0}$ |  $\omega_{2}^{2}$
---|---|---|---|---|---|---|---
|  $[\,\nicefrac{\mathrm{pm}}{\mathrm{s^{2}}}]$ |  $[\,\nicefrac{\mathrm{nrad}}{\mathrm{s^{2}}}]$ |  $[\,\nicefrac{\mathrm{nrad}}{\mathrm{s^{2}}}]$ |  (2016) |  $[\mathrm{h}]$ |  $[\mathrm{nm}]$ |  $[\nicefrac{\mathrm{1}}{\mathrm{s^{2}}}\times 10^{-7}]$
_Nominal_ |  0 (1100) |  15 |  15 |  09/06 |  $18$ |  $\sim 40$ |  $-7.12\pm 0.03$
URLA |  0 (26) |  2.2 |  1.4 |  18/12 |  $132$ |  $\sim 30$ |  $-4.53\pm 0.09$
Table 1: Free-fall mode measurement data set. The table includes the authority
levels, duration ($\Delta t$, in hours), initial flight amplitude ($\Delta
x_{0}$) and TM2 stiffness ($\omega_{2}^{2}$). In both cases $\omega_{2}^{2}$
is in agreement (1$\sigma$) with the values resulting from the system dynamics
calibration presented in Ref. Armano  _et al._ (2018b). The force authority
values in bracket refer to the continuous control.

_Data analysis.—_ The analysis of the free-fall mode experiment is challenging
due to the presence of impulses. Estimating the noise in $\Delta g$ without
actuation implies limiting the analysis to the free-fall periods alone,
effectively “gapping” data to be insensitive to the noise from the high-force
impulses. The effect of gaps on the spectrum must be characterized, especially
at low frequency, where the noise is expected to be lower than in presence of
control Weber  _et al._ .

In general, gaps can corrupt the spectral estimation, in the form of spectral
leakage from both high and low frequencies, thus introducing a systematic bias
in the underlying spectrum. Gaps can be masked with smooth spectral windows or
filled with synthetic noise. In this Letter we present the results obtained by
applying the “Blackman-Harris gap zero” (BHGZ) technique (see Ref. Russano
_et al._ (2018a) for a full review, except for the bias removal). The method,
implemented using the dedicated data analysis toolbox, LTPDA Hewitson  _et
al._ (2009), consists in filling the gaps with zero numerically by means of a
rectangular-wave window, after having low-pass filtered and decimated the
$\Delta g$ time series. The name of the approach refers to the shape of the
filter chosen, that is a minimum 4-term Blackman-Harris (BH) window. The
filter is applied to reduce the aliasing caused by the rectangular-wave window
and it is a finite impulse response (FIR) filter to avoid mixing in the gaps.
Indeed, compared to smoother windows, the rectangular-wave window produces a
relevant spectral leakage of the noise, from high frequency into the low
frequency band of the spectrum. Finally, the downsampling is imposed by the
numerical limitation of the procedure applied to remove, from the spectrum,
the remaining bias due to gaps. This procedure will be described below, while
implementation details of the BHGZ technique are found in Refs. SM (1);
Giusteri  _et al._ (2017).

The PSD of filtered, decimated and gapped data is estimated with the same
technique as for the continuous data described in Ref. Armano  _et al._
(2016a), with errors estimation based on $\chi^{2}$ statistics Armano  _et
al._ (2018a). Then, it is normalized for the transfer function of the BH
filter and finally corrected for the bias induced by gaps.

The spectral bias in free-fall data appears in the form of peaks at harmonics
of the gap frequency ($\equiv 1/T_{\rm{exp}}$ $\sim$ 2.8 mHz, see Fig. 3),
observed after the multiplication of data by the rectangular-wave window. In
addition, the amplitude of the gapped spectrum is reduced compared to that of
continuous data, due to removal of data points, as reported e.g., in Ref.
Baghi  _et al._ (2015) and discussed also in the Supplemental Material SM (1),
and this reduction scales with the gap size. In particular, in case of white
noise, the normalization factor needed to compensate for the missing points
set to zero, is equal to the inverse of the rectangular-wave duty cycle, as
demonstrated in Ref. Russano  _et al._ (2018a).

To remove the bias we follow “a pseudo-inverse” approach, described in detail
in the Supplemental Material SM (1), based on looking for the theoretical
shape of the spectrum that, through the action of the rectangular-wave window,
reproduces the experimental spectrum. In practice, we fit the gapped spectrum
to a smooth continuous model, we assume underneath data, which is convolved
with the rectangular-wave window. In our case, the low-frequency noise only is
modeled and the fit is performed at samples away from the peaks which we do
not model. Indeed, we are mainly interested in removing the bias at low
frequencies in order to estimate the noise in absence of the compensation
force on TM2. The noise model, reported in Eq. (7) in the Supplemental
Material SM (1), is based on the measured noise with continuous actuation
Armano  _et al._ (2018a) and it is precise enough as we achieve a good quality
of fit [see, as an example, Fig. 2(b) in the Supplemental Material SM (1)].

The fit parameters are then used to trace the “native” spectrum of free-fall
mode data without gaps, as explained in the Supplemental Material SM (1).
Figure 3 shows the result, in terms of $\Delta g$ ASD (amplitude spectral
density), of this procedure on data of the free-fall mode experiment carried
out in December. The result obtained from the best fit to the ASD of data
(solid line) is indicated by the dashed line, while the dash-dotted line is
the model for the underlying continuous differential acceleration noise
spectrum, resulting from our analysis, which converts into the dashed line
when gaps are inserted. Thus, the bias is removed from the experimental gapped
spectrum (solid line), by multiplying it by the ratio between the dash-dotted
and the dashed lines. The effective experimental curve, with points
appropriately scaled by the ratio of the dash-dotted and dashed lines, is
shown by the dot data points in Fig. 4. Details of the analysis of December
data, can be found in the Supplemental Material SM (1).

Applying the technique on continuous control $\Delta g$ data, with
artificially inserted gaps, accurately recovers the spectrum obtained when
analyzing the full continuous data set. The results of the method calibration
are reported in the Supplemental Material SM (1).

Figure 3: Fit results to free-fall data, measured in December 2016, to remove
the bias from the spectrum (see the text for details). The peaks are excluded
from the analysis and hence not fitted.

_Results.—_ _URLA authority:_ Figure 4 shows the $\Delta g$ ASD of the free-
fall mode experiment performed with URLA $\phi$ authority (asterisk data
points), compared with that measured with continuous control mode in the same
authority and just after the free-fall mode experiment (dot data points).
Figure (4) includes the actuation noise predictions in URLA authority for both
the measurements Weber  _et al._ ; Giusteri  _et al._ (2017), showing that
actuation noise does not dominate the low frequency spectrum in URLA
continuous control mode and that it is expected to lessen, in free-fall mode,
by roughly 20$\%$ at 0.1 mHz in ASD. The shadowed area behind the data points
coincides with that of the dash-dotted line of Fig. 3. As visible, at
frequencies below 1 mHz the $\Delta g$ estimate in URLA free-fall mode agrees,
within 1$\sigma$, with that measured in continuous control. Thus, removing the
$x$ control does not significantly reduce noise along the sensitive axis,
since actuation noise in continuous mode is already dominated by the $\phi$
control, which does not change in free-fall mode.

Figure 4: Acceleration noise estimate with free-fall mode in URLA $\phi$
authority compared with that in continuous actuation mode in URLA and the LISA
requirements Amaro-Seoane  _et al._ . The free-fall ASD (dots) results from 20
periodograms, while the continuous noise run (asterisks) is $\sim\,$18 days
long (78 periodograms), both calculated at 1$\sigma$ confidence interval and
according to the method presented in Ref. Armano  _et al._ (2018a). Shadowed
area: estimate of the “native” free-fall spectrum. Dashed lines: actuation
noise predictions.

While the noise reduction is not resolvable, the free-fall mode result
represents an important confirmation of the LPF differential acceleration
benchmark without applied forces. It also confirms that the low frequency
noise excess, visible around 0.1 mHz and currently under investigation Armano
_et al._ (2018a), is not caused by inaccuracies in the $x$-force subtraction,
as the free-fall mode completely removes such contribution: we can state that
noise from possible errors in the $x$-actuator calibration is below our
detection threshold. To conclude, the free-fall mode experiment in the low
$\phi$ authority confirms, as an independent measurement, the LPF performance
achieved in continuous control mode.

_Nominal authority:_ The results of the one-day experiment executed with free-
fall mode in _nominal_ $\phi$ authority, is depicted in Fig. 5 (thick solid
line). The thin solid line indicates a $\Delta g$ estimate measured in the
period of the free-fall run with _nominal_ continuous control. The picture
includes the expected low-frequency noise at that period of time for both the
measurements (dashed lines) Weber  _et al._ . As visible, in this case turning
off the _nominal_ authority ($\sim$ 1100 $\rm{pm}/\rm{s}^{2}$) $x$ actuator,
reduces noise at low frequency effectively, matching the predictions of
suppression of actuation noise along the sensitive axis, which in turn
dominates the spectrum when active. The free-fall mode thus can be considered
an alternative technique to eliminate actuation noise when this is a limiting
factor.

Figure 5: Comparison of $\Delta g$ ASD between the free fall mode experiment
executed in June 2016 in _nominal_ $\phi$ authority (thick solid line, 2
periodograms) and the continuous control mode run carried out in May 2016 in
the same authority (thin solid line, 3 periodograms). The dashed lines are the
noise predictions as explained in the text.

To conclude, though the noise due to the $x$ control does not dominate the
low-frequency band in the low authority scheme, as confirmed by the free-fall
mode results, actuation noise enters in the LISA noise budget through the
$\phi$ control. In this context, the free-fall mode experiment has provided an
acceleration noise measurement in an actuation configuration similar to that
of LISA.

###### Acknowledgements.

_Acknowledgements.—_ This work has been made possible by the LISA Pathfinder
mission, which is part of the space-science program of the European Space
Agency. The French contribution has been supported by CNES (Accord Specific de
projet CNES 1316634/CNRS 103747), the CNRS, the Observatoire de Paris and the
University Paris-Diderot. E. Plagnol and H. Inchauspé would also like to
acknowledge the financial support of the UnivEarthS Labex program at Sorbonne
Paris Cité (ANR: ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). The Albert-
Einstein-Institut acknowledges the support of the German Space Agency, DLR.
The work is supported by the Federal Ministry for Economic Affairs and Energy
based on a resolution of the German Bundestag (FKZ 50OQ0501 and FKZ 50OQ1601).
The Italian contribution has been supported by Agenzia Spaziale Italiana and
Istituto Nazionale di Fisica Nucleare. The Spanish contribution has been
supported by Contracts No. AYA2010-15709 (MICINN), No. ESP2013-47637-P, and
No. ESP2015-67234-P and No. ESP2017-90084-P (MINECO), and 2017-SGR-1469
(AGAUR). M. Nofrarias acknowledges support from Fundacion General CSIC
(Programa ComFuturo). F. Rivas acknowledges support from a Formación de
Personal Investigador (MINECO) contract. The Swiss contribution acknowledges
the support of the Swiss Space Office (SSO) via the PRODEX Programme of ESA.
L. F. acknowledges the support of the Swiss National Science Foundation
Project Number 200021-162449. The UK groups wish to acknowledge support from
the United Kingdom Space Agency (UKSA), the University of Glasgow, the
University of Birmingham, Imperial College, and the Scottish Universities
Physics Alliance (SUPA). J. I. Thorpe and J. Slutsky acknowledge the support
of the U.S. National Aeronautics and Space Administration (NASA).

## References

  * McNamara  _et al._ (2008) P. McNamara, S. Vitale, K. Danzmann, and on (LISA Pathfinder Science Working Team), Classical and Quantum Gravity 25, 114034 (2008).
  * (2) P. Amaro-Seoane _et al._ , ArXiv: 1702.00786 .
  * Armano  _et al._ (2016a) M. Armano _et al._ , Phys. Rev. Lett. 116, 231101 (2016a).
  * Antonucci  _et al._ (2011) F. Antonucci _et al._ , Classical and Quantum Gravity 28, 094002 (2011).
  * Grynagier  _et al._ (2009) A. Grynagier, W. Fichter, and S. Vitale, Classical and Quantum Gravity 26, 094007 (2009).
  * Rummel  _et al._ (2011) R. Rummel, W. Yi, and C. Stummer, C J Geod 85, 777 (2011).
  * Carbone  _et al._ (2005) L. Carbone, A. Cavalleri, R. Dolesi, C. D. Hoyle, M. Hueller, S. Vitale, and W. J. Weber, Classical and Quantum Gravity 22, S509 (2005).
  * Dolesi  _et al._ (2003) R. Dolesi, D. Bortoluzzi, P. Bosetti, L. Carbone, A. Cavalleri, I. Cristofolini, M. DaLio, G. Fontana, V. Fontanari, B. Foulon, C. D. Hoyle, M. Hueller, F. Nappo, P. Sarra, D. N. A. Shaul, T. Sumner, W. J. Weber, and S. Vitale, Classical and Quantum Gravity 20, S99 (2003).
  * Armano  _et al._ (2016b) M. Armano _et al._ , Classical and Quantum Gravity 33, 235015 (2016b).
  * Armano  _et al._ (2018a) M. Armano _et al._ , Phys. Rev. Lett. 120, 061101 (2018a).
  * (11) W. J. Weber _et al._ , (to be published) .
  * Giusteri  _et al._ (2017) R. Giusteri _et al._ , J. Phys. Conf. Ser. 840, 012005 (2017).
  * Armano  _et al._ (2018b) M. Armano _et al._ , Phys. Rev. D 97, 122002 (2018b).
  * Russano  _et al._ (2018a) G. Russano, A. Cavalleri, A. Cesarini, R. Dolesi, V. Ferroni, F. Gibert, R. Giusteri, M. Hueller, L. Liu, P. Pivato, H. B. Tu, D. Vetrugno, S. Vitale, and W. J. Weber, Classical and Quantum Gravity 35, 035017 (2018a).
  * Hewitson  _et al._ (2009) M. Hewitson _et al._ , Classical and Quantum Gravity 26, 094003 (2009).
  * SM (1) See Supplemental Material at , sections “BHGZ method implementation” for implementation details of the BHGZ technique, “Final considerations” for discussion about the amplitude of the spectrum peaks, “Bias removal algorithm” for details on the bias removal algorithm, “Bias removal on free-fall mode data measured in December” for the bias removal procedure applied on data of free-fall mode, “Calibration of the method with continuous noise data” for the bias removal procedure applied on continuous data for calibration purposes, which include Refs. Armano  _et al._ (2018a); Russano  _et al._ (2018b); Armano  _et al._ (2018a); Vitale  _et al._ (2014); Ferraioli  _et al._ (2011); Armano  _et al._ (2018c); Welch (1967).
  * Baghi  _et al._ (2015) Q. Baghi, G. Métris, J. Bergé, B. Christophe, P. Touboul, and M. Rodrigues, Phys. Rev. D 91, 062003 (2015).
  * Welch (1967) P. Welch, IEEE Trans. Audio and Electroacoust. 15, 70 (1967).
  * Russano  _et al._ (2018b) G. Russano, A. Cavalleri, A. Cesarini, R. Dolesi, V. Ferroni, F. Gibert, R. Giusteri, M. Hueller, L. Liu, P. Pivato, H. B. Tu, D. Vetrugno, S. Vitale, and W. J. Weber, Classical and Quantum Gravity 35, 035017 (2018b).
  * Vitale  _et al._ (2014) S. Vitale _et al._ , Phys. Rev. D 90, 042003 (2014).
  * Ferraioli  _et al._ (2011) L. Ferraioli, G. Congedo, M. Hueller, S. Vitale, M. Hewitson, M. Nofrarias, and M. Armano, Phys. Rev. D 84, 122003 (2011).
  * Armano  _et al._ (2018c) M. Armano _et al._ , Supplemental Material, Phys. Rev. Lett. 120, 061101 (2018c), section on Bayesian ASD estimate.