# DMS: Differentiable Mean Shift for Dataset Agnostic Task Specific Clustering
Using Side Information

Michael Hobley, Victor Prisacariu
Active Vision Laboratory
University of Oxford
[mahobley<EMAIL_ADDRESS>

###### Abstract

We present a novel approach, in which we learn to cluster data directly from
side information, in the form of a small set of pairwise examples. Unlike
previous methods, with or without side information, we do not need to know the
number of clusters, their centers or any kind of distance metric for
similarity. Our method is able to divide the same data points in various ways
dependant on the needs of a specific task, defined by the side information.
Contrastingly, other work generally finds only the intrinsic, most obvious,
clusters. Inspired by the mean shift algorithm, we implement our new
clustering approach using a custom iterative neural network to create
Differentiable Mean Shift (DMS), a state of the art, dataset agnostic,
clustering method. We found that it was possible to train a strong cluster
definition without enforcing a constraint that each cluster must be presented
during training. DMS outperforms current methods in both the intrinsic and
non-intrinsic dataset tasks.

Figure 1: Examples of learnt clusters given side information. Positive
examples contain two data points known to be from the same class and negative
examples are from different classes, here the same class is either the same
digit (top) or same person (bottom) and negative examples are of two different
classes. Our method learns to correctly identify unseen classes, for example
the 7, 8, 9 clusters are not present when training on MNIST (top).

##  Introduction

Clustering algorithms have been extensively studied as they are a ubiquitous
tool. Classical clustering approaches normally require no known ground truth
class labels but do need either a known number of clusters as in $k$-means [1]
or DBSCAN [2], or a predefined similarity distance tolerance, as in mean shift
[3]. In many cases it is difficult to specify an informative distance or to
know the number of clusters in the dataset. Generally deep learning based
methods use these classical algorithms either to supervise their training [4,
5] or to directly cluster a projected feature space [6, 7]. These approaches
suffer from the same problems as the classical methods, and require parameter
tuning. Another, often ignored difficulty is the fact that many datasets could
be clustered in multiple valid ways, as seen in Figure 4, each useful for a
different end task.

Our approach is inspired by other contemporary work [8, 9, 10] that uses
pairwise similarity based side information to cluster a dataset, and by mean
shift which iteratively finds clusters using a distance metric as a proxy for
similarity; thus, we constructed Differentiable Mean Shift (DMS). DMS is the
natural progression of these two approaches as it removes the proxy and trains
a similarity kernel to find clusters directly from only side information. As
in other contemporary work [8, 9, 10], the side information takes the form of
pairwise ’similar/dissimilar’ labels. With DMS we iteratively train a kernel
to evaluate the similarity of a given pair of points in the context of a
specific task, thus DMS can cluster the same dataset in multiple valid ways by
altering the side information. Unlike other classical or deep clustering
methods [4, 5, 6, 11], this learnt definition shows promise without the need
for: i) wider knowledge of the dataset (for example, the number of clusters,
their centers or a distance metric), ii) all classes to be present or evenly
represented in the example data iii) an embedded feature space or an explicit
dimensional reduction stage, iv) hyper-parameter tuning for different datasets
or tasks, which are present in other methods [4, 5, 6, 11, 12, 13, 14, 15]. As
we can learn a task specific similarity kernel without all classes present in
the side information DMS is able to identify novel, unseen classes at
inference time as seen in Figure 1. As our method operates as a single forward
pass through a lightweight network which does not require tuning, dimensional
reduction or an understanding of the whole dataset, it is trivial to add it
into another deep learning method either as a head or for end-to-end training.

Our contributions are (i) a new approach to clustering, which directly
clusters points trained using only side information rather than using distance
tolerances or a number of clusters (ii) a differentiable, mean shift inspired
framework that can be easily added to other deep networks (iii) a state of the
art algorithm that outperforms other contemporary methods on intrinsic tasks
and is capable of performing non-intrinsic task clustering on various
datasets.

The remainder of this paper is structured as follows. Section 2 briefly
discusses classical clustering algorithms before discussing current deep
learning based approaches to clustering data. Section 3 outlines and
formalises the mathematics of the classical mean shift algorithm, and our
differentiable mean shift process. Section 4 discusses the details of our
method and evaluates it in the context of other recent clustering approaches.

##  Related Work

Here we consider classical approaches to clustering before discussing recent
deep learning based approaches.

###  Classical Clustering

Classic deterministic clustering algorithms can be roughly grouped into
spatial clustering [1, 2] and spectral clustering [16].

Spatial clustering methods use a metric function, typically the L1 norm,
Euclidean norm or cosine distance, to evaluate the difference or similarity
between two data points. For instance $k$-means [1], minimises the sum of the
distances between each data point and its closest cluster centers. $K$-means++
[17] improves the initialisation by finding spread out sample points, $k$
points are randomly selected, but the selection is weighted based on the
squared distance between it and other already sampled points. Another
improvement on $k$-means came from Xing et al. [8], who focused on learning a
more appropriate distance metric using pairwise side information. This idea
has since been extensively explored [14, 18, 19, 20, 21]. $K$-means and its
variants benefit from being both simple and robust, however, in practice, $k$
is often unknown.

Another branch of spatial methods adopts a bottom up strategy, in which small
clusters are merged using a metric function until only one cluster is left.
Examples are the Ward’s hierarchical clustering (AC-W) [22] and DBSCAN [2].
DBSCAN repeatedly expands a cluster from a seed if the density is above a
threshold area. The GDL [23] agglomeratively merges sub clusters according to
a metric function that takes into account local geometry using in/out degrees
of a data point.

Spectral clustering methods [16, 24, 25] convert a data point graph into the
eigenspace of its laplacian. Given $k$ is equal to the multiplicity of the
eigenvalues and clusters can be retrieved from the eigenvectors, the
eigenspace can be used to improve the $k$-means initialisation.

Classic clustering approaches use simple metric functions and work well with
low-dimension data. However, they generalise poorly in higher dimension space.
This is known as the curse of dimensionality [26, 27, 28].

###  Deep Clustering

Current deep learning methods attempt to either (i) project features into a
space that can be clustered using classical methods effectively (ii) mimic
conventional spatial/spectral clustering approaches with a set of trainable
layers [29].

For example, Deep Clustering Network (DCN) [7] uses a regularised $k$-means
loss. JULE [30] uses an agglomerative clustering loss to learn feature
embeddings. Deep Embedding for Clustering (DEC) [31] and DEPICT [11] both use
an assignment loss which measures the KL divergence between their predictions
and a soft assignment. Robust Continuous Clustering (RCC) [4] and Deep
Continuous Clustering (DCC) [5] use a locality-preserving loss to learn a
robustified feature embedding that is consistent with a classical $k$-NN
graph. DAC [6] iteratively improves the feature space driving similar points
together using binary pairwise-classification loss.

In order to tackle the problem posed by high dimensional spaces, many methods
have applied deep learning to project the features into a lower dimensional
space. For instance, RCC-DR [4] and DCN both train a linear autoencoder for
dimension reduction, DEPICT and DCC both improve upon previous works by adding
deep autoencoders. While autoencoders are the most commonly used dimensional
reduction approach, other methods have been proposed to similar effect, for
example Hu et al. [32], Hsu and Lin [33], and [34], reduce their feature
spaces using self-augmentation and a GAN respectively.

The impact of an improved feature space is clear, DAC [6] generates a feature
space that can be effectively clustered by $k$-means even for challenging
datasets, however, DAC offers negligible improvements to the clustering
itself. In this work we are focusing on clustering a feature space without
encoding it, as it is not always possible or efficient to train an encoder for
the clustering, for example when the clustering is added as a head. Of course
the above encoding methods could compliment our approach well.

Deep clustering approaches that use classical clustering methods as a base
have the same flaws as those classical methods i.e. they require some
supervision in the form of dataset-specific parameters. This parameter tuning
significantly decreases the usefulness of these clustering approaches for end-
to-end training, as well as for non deep-learning applications.

Recently there have been various works that focus on semi or completely
supervised clustering [12, 35, 36, 37]. These approaches however have
generally focused on learning a feature encoding optimised for clustering
rather than focusing on creating a robust clustering method to work on any
features. As discussed above, these works could be used to compliment our work
but focus on a fundamentally different aspect of the problem.

Following Xing et al. [8], the use of pairwise side information for deep
clustering has been further explored as an alternative to direct supervision
[9, 13, 12, 15, 38]. However, these approaches have similar drawbacks to
previous clustering methods. They require a knowledge of the data beyond
solely pairwise examples, commonly the number of ground truth clusters present
in the dataset [9, 12, 13].

##  Mean Shift

Given a set of data points, mean shift [3] finds the appropriate cluster
centers and compares these in order to cluster the data points. Cluster
centers are found by iteratively averaging the position of points currently
believed to be inliers to the same cluster. In classical mean shift this is
based around two assumptions: firstly, points that are similar belong to the
same cluster, and, secondly, that euclidean distance is a good proxy for
similarity. The mean shift algorithm is formulated as follows.

Given a set, $\mathcal{X}$, where $|\mathcal{X}|=M$ and any data point
$\mathbf{x}\in\mathcal{X}$ is a $N$-Dimension vector $\mathbb{R}^{N}$. The
sample mean function,
$m:\mathcal{X}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$, on $\mathcal{X}$
is defined by:

$\bar{\mathbf{x}}^{(n+1)}=m(\mathcal{X},\bar{\mathbf{x}}^{(n)})=\frac{\sum_{\mathbf{x}\in\mathcal{X}}\mathbf{x}K(\mathbf{x},\bar{\mathbf{x}}^{(n)})}{\sum_{\mathbf{x}\in\mathcal{X}}K(\mathbf{x},\bar{\mathbf{x}}^{(n)})},$
(1)

where $\bar{\mathbf{x}}^{(n)}$ is the current sample mean calculated in the
$n$-th iteration, $\mathbf{x}\in\mathcal{X}$ is a data point, and $K(.)$ is
the kernel function which evaluates the similarity of two points,
$K:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow\\{0,1\\}$.

Given the sample mean function $m(.)$ defined on some collection of data with
an initial value $\bar{\mathbf{x}}^{(0)}$, a cluster centre together with the
cluster inlier data points can be calculated using the mean shift algorithm
[3]. The sample mean is iteratively updated using Equation 1 until the
distance of sample means between two consecutive iterations becomes
sufficiently small, i.e.
$\|\bar{\mathbf{x}}^{(n+1)}-\bar{\mathbf{x}}^{(n)}\|<\tau$, where $\tau$ is
the threshold. The difference between the converged sample mean
$\bar{\mathbf{x}}$ and the start point $\bar{\mathbf{x}}^{(0)}$ is known as
the mean shift.

###  Differentiable Kernel

The original Mean shift algorithm relies on a kernel function $K_{F}(.)$
measuring the similarity between two data points using their Euclidean
distance, which is known as the flat kernel, and is not differentiable. Any
data point within the $\lambda\text{-ball}\subset\mathbb{R}^{N}$ centred at
$\bar{\mathbf{x}}$ of the cluster is considered as an inlier.

$K_{F}(\mathbf{x}_{1},\mathbf{x}_{2})=\begin{cases}1&\text{if}\
\|\mathbf{x}_{1}-\mathbf{x}_{2}\|_{2}\leqslant\lambda\\\ 0&\text{if}\
\|\mathbf{x}_{1}-\mathbf{x}_{2}\|_{2}\geqslant\lambda\end{cases}$ (2)

Another commonly used similarity kernel is the unit Gaussian Kernel [39]. The
Gaussian Kernel has no neighbourhood characteristic, including all points in
$\mathcal{X}$, but scales the importance of points based on their distance to
the sample mean. While the Gaussian Kernel is differentiable, its
representation capability is limited.

Instead of using distance as a proxy for similarity we propose a
differentiable kernel based on a neural network,
$K_{D}:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow[0,1]$, that learns
similarity directly. The differentiable kernel, $K_{D}(.)$ can produce a
probability to measure the similarity between two data points or a data point
and the sample mean $\bar{\mathbf{x}}$. Particularly, the more similar, the
output of $K_{D}(.)$ is closer to 1. Due to the flexibility of the neural
network, $K_{D}(.)$ can be much more descriptive than the classical analytical
metrics, such as the aforementioned flat or Gaussian kernels.

In high dimensional space it is desirable to be able to act non-symmetrically
towards a dataset’s dimensions based on a task’s requirements. A neural
network is ideal for this as each dimension can be acted on independently to
find those most salient to a given task. In contrast simpler metrics like the
Euclidean norm are not able to apply the same contextual knowledge of the
feature space, and always evaluate dimensions with greater absolute variation
as more significant. This neural network ability decreases the need for
explicit dimensional reduction and removes the problems of poor representation
inherent in such a reduction. The context based dimensional independence also
allows a single feature space to be clustered differently dependent on task.
The specific architectures of our differentiable kernels can be seen in Table
1. The output of $K_{D}$ is bounded to $[0,1]$ by a sigmoid function. This
sigmoid also introduces a non-linear scaling similar to the traditional
Gaussian kernel, this scaling increases and decreases the impacts of the areas
close to and far from the sample mean respectively.

TABLE 1: Differential subtraction and concatenation based kernel
architectures, as described in Section 4.4.

Inputs | Sample Mean, $\bar{\mathbf{x}}_{i}$ | N
---|---|---
Data point, $\mathbf{x}$ | N

$K_{D}(\mathbf{x},\bar{\mathbf{x}}_{i})$ $subtract$ | Subtract, $(\mathbf{x}-\bar{\mathbf{x}}_{i})$ | N
---|---|---
| $\begin{bmatrix}\text{fully connected,}&N/2\\\ \text{fully
connected,}&N/4\\\ \text{fully connected,}&1\end{bmatrix}$
---
Sigmoid
| $\begin{matrix}N/2\\\ N/4\\\ 1\\\ 1\\\ \end{matrix}$
---
$K_{D}(\mathbf{x},\bar{\mathbf{x}}_{i})$ $concat$ | Concatenate, $concat(\mathbf{x},\bar{\mathbf{x}}_{i})$ | 2 $\times$ N
| $\begin{bmatrix}\text{convolution,}&2\times 1,N\\\ \text{fully
connected,}&N/2\\\ \text{fully connected,}&N/4\\\ \text{fully
connected,}&1\end{bmatrix}$
---
Sigmoid
| $\begin{matrix}N/2\\\ N/4\\\ 1\\\ 1\\\ 1\\\ \end{matrix}$
---

###  Differentiable Mean Shift

Our method, which in practicality is a single pass through a neural network,
is comprised of two stages, iterative center finding and refined inlier
prediction, as shown in Figure 2. During the first stage the cluster centers
are found by iteratively taking the weighted average of their inliers, using
their predictions, $K_{D}(\mathbf{x}_{i},\bar{\mathbf{x}})$, as the weighting.
In the second stage the confidence of each point being an inlier to each
cluster is used to combine similar centers, this stage also finds the true
cluster each point is associated with. The second stage can also be thought of
as the same as a final iteration of the first stage; it uses the same network
and weights to find inlier predictions, but does not subsequently use these
predictions as weights to find a center.

Data Points

Center Finding Stage

Inlier Prediction Stage

Figure 2: Algorithm Stages. First, the centers are located with an initial
approximation of the clusters. Then, given the centers inliers are then found,
and the clusters are refined, similar clusters are combined, subset clusters
are incorporated and noise points are removed.

To estimate the sample mean vector of one cluster, we initialise
$\bar{\mathbf{x}}^{(0)}=\mathbf{x}_{i}$, where $\mathbf{x}_{i}\in\mathcal{X}$,
we then predict the sample mean iteratively,
$\bar{\mathbf{x}}^{(n+1)}=m(\mathcal{X},\bar{\mathbf{x}}^{(n)})$, where $n$ is
the iteration step index, one step can be seen in Figure 3.

Unlike classical mean shift, we do not determine convergence by defining a
distance tolerance between the centers found in consecutive iterations.
Instead, we use the prediction of inliers given the current and previous
sample mean. If two consecutive sample means, $\bar{\mathbf{x}}^{(n-1)}$ and
$\bar{\mathbf{x}}^{(n)}$ predict the same points to be inliers and outliers,
the algorithm has converged and the cluster center has been found.

Figure 3: The DMS framework for cluster prediction.

In this way, we can calculate $M$ sample means, $\bar{\mathbf{x}}_{i}$,
initialised from $\mathbf{x}_{i}$, where $i=1,..,M$.

We evaluate sample mean similarity in the same way as we find our iterative
convergence. This is due to the fact that while we observe that various
samples, $\mathbf{x}_{i}$, may robustly converge to an identical cluster
center, i.e. $\|\bar{\mathbf{x}}_{i}-\bar{\mathbf{x}}_{j}\|_{2}<\epsilon$
where $\epsilon$ is an infinitesimally small value; we also observe that many
samples that belong to the same ground truth cluster may result in different
but similar sample means. We define two centers to be the same if they predict
the same points to be inliers and outliers with similar confidences.

Formally, we calculate a confidence matrix
$\mathbf{H}\in\mathbb{[}0,1]^{M\times M}$ where the element $\mathbf{H}_{r,c}$
at the $r^{th}$ row and the $c^{th}$ column represents the confidence that the
$c^{th}$ data point belongs to the $r^{th}$ candidate cluster
$\mathbf{H}_{r,c}=K_{D}(\mathbf{x}_{c},\bar{\mathbf{x}}_{r})$.

A similarity matrix over the candidate sample means can be calculated
$\mathbf{S}=\tilde{\mathbf{H}}\tilde{\mathbf{H}}^{\top}$, where
$\tilde{\mathbf{H}}$ is the bi-stochastic normalisation of $\mathbf{H}$. Each
element $\mathbf{S}_{i,j}$ measures the similarity between the $i^{th}$ and
$j^{th}$ sample means. The final unique cluster centers can be found by
binarising $\mathbf{S}$ and extracting connected components. The same
binarising tolerance of $0.5$ is used for all datasets. However, we found that
any value between about $0.1$ and $0.95$ will provide similar results as the
values are driven to either $1$ or $0$ by the training loss, see Equation 3.
Each data point is then assigned to the cluster with the highest inlier
confidence for that point.

It may occur that a point that originally converged to a particular center is
not assigned to the associated cluster during the final stage either because a
different center is more confident of its inliership or because no cluster is
confident of its inliership. This is generally a positive effect as points not
associated with any cluster or on the border of two clusters might converge to
an incorrect center during our first stage, due to the lack of a hard
neighbourhood constraint in $K_{D}$. The clusters could then be further
refined in various ways. For example, trivial solutions could be removed if
you know this case is not found in the data. we do not make this assumption
during our testing.

It is worth noting that unlike classical mean shift our algorithm does not
need to be initialised from every point in the dataset. Classical mean shift
classifies a point as belonging to a cluster if it converges to the associated
center; our algorithm only needs to be initialised from one point per cluster,
this point locates the cluster’s center, our second stage then finds all
inliers of this cluster.

###  Training

During training we generate data as follows. A given dataset contains a set of
categories, $\mathcal{C}$, and each category $c\in\mathcal{C}$ contains a
number of samples, $|c|$, this cardinality may vary between categories. One
training instance is composed of: a randomly picked data point called the
initialisation point, $(c^{+},i)$, where $c^{+}\in\mathcal{C}$ and $i$ is the
index of the initialisation point within the category, a set of positive
samples, $\mathcal{P}$, a set of negative samples, $\mathcal{N}$, and a set of
unlabelled samples, $\mathcal{U}$. We use the sets $\mathcal{P}$ and
$\mathcal{N}$ to define our side information labels of ‘similar’ and
‘dissimilar’ respectively.

The category of the initialisation point, $c^{+}$, is named the positive
category. The positive samples are randomly drawn from the points who’s side
information indicates a similarity with the initialisation point, i.e.
$\mathcal{P}=\\{(c^{+},j)\\}$. The negative samples are selected from the rest
of the points with side information, i.e. $\mathcal{N}=\\{(c^{-},i)|c^{-}\neq
c^{+}\cap c^{-}\in\mathcal{C}\\}$, and the unlabelled samples $\mathcal{U}$
are drawn from all points with and without side information,
$\mathcal{U}=\\{(c,i)|c\in\mathcal{C}\\}$.

The specific class label, $c^{+}$, is only used to creating the side
information in the form of pairwise relational labels. The use of side
information instead of the specific class label during training allows the
clustering network to learn a class independent representation of similarity.

The initialisation point, $\mathcal{P}$ and $\mathcal{N}$ are all taken from a
training set with side information, a small randomly drawn subset of the
complete dataset. All points in the training set were excluded during testing.
The unlabelled samples, $\mathcal{U}$, are taken from the entire dataset, and
are excluded from the training loss. The unlabelled data points are only added
to increase the cardinality and variance of the clusters during training. All
classes and data points in $\mathcal{P}$, $\mathcal{N}$ and $\mathcal{U}$ are
randomly and independently selected. Points can be repeated in a training
instance, this allows for training with data where there are not enough
available data points to create the correct positive class size.

During training we use four mean shift iterations, the impact of this is
further discussed in Section 4.4. We used the Binary Cross Entropy of the
inlier predictions and the side information of the labelled points as the
training loss, see Equation 3. Predictions of the unlabelled points are not
used in the loss.

$loss=-\sum_{i\in{\mathcal{P},\,\mathcal{N}}}(y_{i}\log(p_{i})+(1-y_{i})\log(1-p_{i}))$
(3)

where $y_{i}=1$ if $x_{i}\in\mathcal{P}$ and $y_{i}=0$ if
$x_{i}\in\mathcal{N}$, and $p_{i}$ is the prediction confidence that $x_{i}$
is in $\mathcal{P}$. This loss consistently pushes predictions to the extremes
of the unit interval, with values between $0.05$ and $0.95$ rarely predicted
after training.

##  Experiments

In this section we evaluate DMS using standard datasets and metrics for both
intrinsic and non-intrinsic tasks, before discussing our architectural and
training decision.

###  Datasets and Metrics

We draw datasets from various domains, including: object detection, facial
recognition, handwriting comprehension and text classification based on word
frequency. To create a fair comparison, we used the standard benchmark
descriptors and reduced datasets where appropriate, as in the experimental
section of [5]. Specifically, we show results of experiments conducted on
MNIST [40], COIL100 [41], YouTube Faces (YTF) [42] and RCV1 [43].

We report results using the standard metrics: clustering accuracy (ACC) [44]
and Normalised Mutual Information (NMI) [45]. Vinh et al. [46] reported that
NMI is biased towards the generation of finer grained clusters, and that ACC
is unreliable in datasets where there is significant variation in cluster
cardinality. For this reason we also include Adjusted Mutual Information (AMI)
[46] in all results tables. ACC, NMI and AMI are defined by Equation 4, 5 and
6 respectively, and are all in the range $[0,1]$, where $1$ indicates ideal
performance.

$\text{ACC}(\mathbf{l},\hat{\mathbf{l}})=\max_{map}\frac{\sum_{i\in{M}}\delta(\hat{l}_{i},map(l_{i}))}{M}$
(4)

where $\mathbf{l}$ and $\hat{\mathbf{l}}$ are the predictions and ground truth
labels, respectively, $\delta(x,y)=1$ if $x=y$, ($\delta(x,y)=0$, otherwise),
and $map$ ranges over all one-to-one mappings between predicted and ground
truth labels, efficiently found through use of the Hungarian algorithm [47].

$\text{NMI}(\mathbf{l},\hat{\mathbf{l}})=\frac{I(\mathbf{l},\hat{\mathbf{l}})}{\sqrt{H(\mathbf{l})H(\hat{\mathbf{l}})}}$
(5)
$\text{AMI}(\mathbf{l},\hat{\mathbf{l}})=\frac{I(\mathbf{l},\hat{\mathbf{l}})-E[I(\mathbf{l,\hat{l}})]}{\sqrt{H(\mathbf{l})H(\hat{\mathbf{l}})}-E[I(\mathbf{l},\mathbf{\hat{l}})]}$
(6)

where $I(\cdot,\cdot)$ is the mutual information between its inputs,
$H(\cdot)$ is the entropy and $E[\cdot]$ is the expectation function.

###  Experiment Training

During training, 5% of the total dataset had pairwise side information. The
total number of data points used in a training instance,
$|\mathcal{P}|+|\mathcal{N}|+|\mathcal{U}|$, is a random integer between 200
and 300 and varies between different batches. The ratio of points with to
without side information in a training instance,
$|\mathcal{P}\cup\mathcal{N}|:|\mathcal{U}|$, varies between $[\frac{1}{3},1]$
according to the uniform distribution; the ratio of positively to negative
pairs, $|\mathcal{P}|:|\mathcal{P}\cup\mathcal{N}|$, varies between
$[\frac{1}{20},\frac{1}{10}]$ according to the uniform distribution. For
datasets with 10 or fewer classes with similar cardinalities this means the
positive set is under-represented during training, and for classes with more
than 20 classes, the positive set is over-represented. This did not appear to
have any significant effect on performance. Each batch is comprised of 96
training instances.

We found that for datasets where clusters were similar, i.e. cardinality,
inter-cluster variance and intra-cluster variance were comparable, a
significantly smaller amount of training data could be used with no detriment
to the performance. COIL100 for instance could be trained with side
information present for only 0.7% of the data, approximately one point per
class, from each of 25 of the total 100 clusters, without significant
performance differences to using the entire training set. This shows the
networks ability to generalise and identify unseen classes. Conversely, when
there are few and unique clusters with regard to cardinality and variance it
is important that each cluster is well represented in the training data.

###  Results

In the ideal case we would initialise our method from all data points in the
test set at inference time. However we found that as long it was likely a
point was initialised in each cluster there was no detriment to performance
from reducing the number of initialisation points. Due to computational
constraints we initialise from 500 data points randomly selected from the test
set. We did not ensure that each ground truth cluster contained an
initialisation point explicitly. We conducted mean shift iterations which
included all points in the test dataset from these initialisation points,
until the centers converged.

We recognise that in many cases there are numerous useful ways to divide a
dataset. Other methods generally only find the most obvious clusters, the
‘intrinsic’ task, while some do allow alteration, this takes the form of
abstract parameter tuning in terms of the number or variance of clusters. In
contrast our method is versatile and unbiased towards any task and
significantly out performs other methods on less obvious, ‘non-intrinsic’,
tasks, as seen in Table 2.

Intrinsic tasks are those which are most obvious given all facets of the data.
In all of our datasets the intrinsic task involves clustering based on a clear
object class, reflecting the way a person would likely cluster the data points
first, seen in the top row of Figure 4. We report our methods performance on
the intrinsic tasks of each dataset in Table 3. We understand that this
comparison is somewhat unfair due to our use of a small amount of side
information, however we include these numbers for context.

Non-intrinsic tasks are the less obvious task, that generally require some
alterations to importance of some facets of a dataset. These tasks can be
either hierarchical or orthogonal to the intrinsic task or somewhere in
between. A hierarchical task is one where multiple complete classes from the
intrinsic task are grouped together to form ‘parent classes’. For example in
COIL100 we test on root object classes and dominant object colour, examples
are shown in the second and third rows of Figure 4. The root object classes
group all object classes into larger ‘parent’ object-type classes, for example
all toy cars are combined. An orthogonal task is one in which every intrinsic
cluster contains one instance from the new task. For example, the images of
cars from COIL100 could be separated based on orientation (agnostic to the
specific object), this is orthogonal to separating the specific objects
(agnostic to the orientation).

As other approaches generally find the divisions associated with the intrinsic
task of a dataset, orthogonal tasks are performed worse than chance, and it
would be unfair to compare these results. We report the performance of DMS and
other contemporary methods on two hierarchical non-intrinsic tasks in Table 2,
we adjusted the hyper-parameters of the benchmark methods to maximise their
chance of success on each of the non-intrinsic tasks.

Figure 4: Different clustering tasks using the same data points. The intrinsic task is the most ‘obvious’ divisions of a dataset this is the task other approaches generally solve. As our method is example lead it has no bias towards the intrinsic task over other problems. TABLE 2: The ability of our method and other contemporary methods to learn various clustering tasks from the same COIL100 features. Both tasks are using the same features as the intrinsic task, shown in Table 3, but focus on a different aspect or scale of the semantic information. | Intrinsic Task | Root Class | Dominant Colour
---|---|---|---
Algorithms | ACC | NMI | AMI | ACC | NMI | AMI | ACC | NMI | AMI
RCC-DR | 0.471 | 0.544 | 0.461 | 0.825 | 0.963 | 0.957 | 0.415 | 0.476 | 0.359
DCC | 0.858 | 0.967 | 0.962 | 0.417 | 0.486 | 0.365 | 0.484 | 0.547 | 0.480
DMS (sub) | 0.990 | 0.996 | 0.994 | 0.990 | 0.997 | 0.996 | 0.817 | 0.790 | 0.751
TABLE 3: The clustering accuracy of our approach compared against 14 baselines on the intrinsic task of each dataset. Accuracy is measured using the standard accuracy metric (ACC), Normalised Mutual Information (NMI), and Adjusted Mutual Information (AMI). Results other than our own are as found in [5]. | MNIST | COIL100 | YTF | RCV1
---|---|---|---|---
Algorithms | ACC | NMI | AMI | ACC | NMI | AMI | ACC | NMI | AMI | ACC | NMI | AMI
K-means++ | 0.532 | 0.5 | 0.5 | 0.621 | 0.835 | 0.803 | 0.624 | 0.788 | 0.783 | 0.529 | 0.355 | 0.355
AC-W | 0.571 | 0.679 | 0.679 | 0.697 | 0.876 | 0.853 | 0.647 | 0.806 | 0.801 | 0.554 | 0.364 | 0.364
DBSCAN | 0.000 | 0.000 | 0.000 | 0.921 | 0.458 | 0.399 | 0.675 | 0.756 | 0.739 | 0.571 | 0.017 | 0.014
SEC | 0.545 | 0.469 | 0.469 | 0.648 | 0.872 | 0.849 | 0.562 | 0.760 | 0.745 | 0.425 | 0.069 | 0.069
LDMGI | 0.723 | 0.761 | 0.761 | 0.763 | 0.906 | 0.888 | 0.332 | 0.532 | 0.518 | 0.667 | 0.382 | 0.382
Deep Learning |  |  |  |  |  |  |  |  |  |  |  |
GDL | n/a | n/a | n/a | 0.825 | 0.965 | 0.958 | 0.497 | 0.664 | 0.655 | 0.444 | 0.020 | 0.020
RCC | 0.876 | 0.893 | 0.893 | 0.831 | 0.963 | 0.957 | 0.484 | 0.85 | 0.836 | 0.356 | 0.138 | 0.138
RCC-DR | 0.698 | 0.827 | 0.828 | 0.825 | 0.963 | 0.957 | 0.579 | 0.882 | 0.874 | 0.676 | 0.442 | 0.442
RCC-DR(SGD) | 0.696 | 0.827 | 0.827 | 0.855 | 0.967 | 0.961 | 0.473 | 0.845 | 0.830 | 0.354 | 0.106 | 0.106
DCN | 0.560 | 0.570 | 0.570 | 0.6002 | 0.830 | 0.810 | 0.620 | 0.810 | 0.790 | 0.730 | 0.470 | 0.470
DEC | 0.867 | 0.853 | 0.840 | 0.815 | 0.645 | 0.611 | 0.643 | 0.811 | 0.807 | 0.683 | 0.504 | 0.500
JULE | 0.800 | 0.900 | 0.900 | 0.911 | 0.983 | 0.979 | 0.342 | 0.587 | 0.574 | - | - | -
DEPICT | 0.968 | 0.919 | 0.919 | 0.402 | 0.678 | 0.667 | 0.586 | 0.790 | 0.785 | - | - | -
DCC | 0.963 | 0.915 | 0.913 | 0.858 | 0.967 | 0.962 | 0.699 | 0.908 | 0.903 | 0.563 | 0.498 | 0.495
DMS ($subtract$) | 0.976 | 0.939 | 0.939 | 0.990 | 0.996 | 0.994 | 0.923 | 0.949 | 0.912 | 0.760 | 0.587 | 0.532
DMS ($concat$) | 0.958 | 0.914 | 0.896 | 0.990 | 0.997 | 0.994 | 0.832 | 0.896 | 0.820 | 0.582 | 0.603 | 0.453

###  Ablation

In this section we evaluate our training method and network architecture. We
test the number of mean shift iterations used during training, the number of
layers used in the kernel and the robustness of subtraction and concatenation
based kernels to limited training data.

Training Iterations. We test the number of iterations used during the training
of our kernel, the results are shown in Table 4. We hypothesise that when a
small number of iterations are used the network does not have the sufficient
flexibility to learn the nuance of the given task, however, with a large
number of iterations the network is not incentivised to effectively define the
clusters as it has many opportunities to improve its prediction. We found that
the number of iterations required for convergence at inference time increased
proportionally with both the number of training iterations and the complexity
of the dataset. This verifies the idea that a single pass through the kernel
is less able to distinguish inliers from outliers given more training
iterations. Given four training iterations the time taken to converge at
inference time varied between five iterations and a single iteration depending
on the complexity of the dataset.

Kernel Depth. We also tested the effect of changing the number of fully
connected layers used in the differentiable kernel, as shown in Table 4. All
networks were trained for the same number of epochs, we used a sufficient
number of epochs to ensure all networks had converged. After three layers the
performance improvements are negligible, while training time and likelihood of
overfitting increase. For this reason we use three fully connected layers for
all other tests.

Concatenation and Subtraction Kernels. We tested two differentiable kernels,
the subtraction based kernel, which takes the difference of the data point and
sample mean, $(\mathbf{x}-\bar{\mathbf{x}}_{i})$, and an alternate,
concatenation based kernel, which takes a concatenation of the data point and
sample mean, $concat(\mathbf{x},\bar{\mathbf{x}}_{i})$. The concatenation
requires the addition of a convolutional layer, with a rectangular $2\times 1$
kernel, before the fully connected layers. We use the same architecture as the
original differential kernel after this convolution. Both architectures are
outlined in Table 1.

We hypothesised the use of a concatenation and a convolutional layer would be
more effective than the simpler subtraction as a greater amount of information
is used. It should be noted that the concatenation and a single convolutional
layer could simply learn the subtraction function. The performances are of
both are shown in Table 3.

We further test the performance of our subtraction and concatenation based
kernels when less side information is available. Here we explicitly limit the
number of classes and points per class with side information, this is unlike
our other experiments where the points with side information are randomly
selected. The results of this experiment appear in Figure 5.

It can be seen that both kernels degrade similarly when fewer data points per
class are seen during training with little performance effect unless the
number of data points is significantly restricted. It can also be seen that
that while the concatenation based kernel performs better when all ground
truth classes are present in the training data, it’s performance is
significantly reduced when this is not the case. The subtraction based kernel
on the other hand, has little degradation in performance when decreasing the
classes present during training, demonstrating its better ability to
generalise to unseen data. This could be due to the fact the subtraction based
kernel, unlike the concatenation kernel is only presented with the relative
position of points to the current sample mean. During training the subtraction
based kernel is unable to use the global position of the training clusters as
a crutch and must learn to find a better description of similarity. This
experiment also clearly shows the methods ability to correctly identify novel
classes.

##  Conclusion

In this paper we have presented a new approach to clustering, one which learns
to cluster points directly from side information in the form of pairwise
similarity. Unlike previous methods, we do not require any other knowledge of
the dataset as a whole; we do not need to know the number of clusters, their
centers nor any kind of distance metric as a prior. We demonstrate this
approach using DMC, a deep clustering method inspired by the mean shift
algorithm.

DMS learns the requirements of a task and uses this to out perform state of
the art deep learning based clustering approaches on both the intrinsic and
non intrinsic tasks without the need to specify any task or dataset specific
hyper-parameters. This flexibility along with DMS’s performance in high
dimensional feature spaces and its lightweight nature enables it to be
seamlessly incorporated into other deep learning applications for end-to-end
training or as a head.

TABLE 4: The performance of our the differentiable kernel with varied training iterations and number of fully connected layers. Both experiments were conducted on the YTF dataset. | Training Iterations | Fully Connected Layers
---|---|---
| 2 | 4 | 6 | 8 | 1 | 2 | 3 | 4
ACC | 0.818 | 0.923 | 0.890 | 0.780 | 0.069 | 0.920 | 0.923 | 0.905
NMI | 0.920 | 0.949 | 0.915 | 0.837 | 0.000 | 0.934 | 0.939 | 0.942
AMI | 0.847 | 0.912 | 0.883 | 0.813 | 0.000 | 0.895 | 0.912 | 0.898
Figure 5: The effect of varying the distribution of training data on the
performance of our subtraction and convolution based kernels. For this test we
explicitly limit the number of classes and the maximum number of points per
class with side information during training. This shows the methods ability to
discover novel classes as there is very little decrease in performance of the
subtraction based method when half of the classes are absent during training.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-053 LHCb-PAPER-2013-011 27 May 2013

Precision measurement of $D$ meson mass differences

The LHCb collaboration†††Authors are listed on the following pages.

Using three- and four-body decays of $D$ mesons produced in semileptonic
$b$-hadron decays, precision measurements of $D$ meson mass differences are
made together with a measurement of the $D^{0}$ mass. The measurements are
based on a dataset corresponding to an integrated luminosity of
$1.0~{}\mbox{\,fb}^{-1}$ collected in $pp$ collisions at 7
$\mathrm{\,Te\kern-1.00006ptV}$. Using the decay $D^{0}\rightarrow
K^{+}K^{-}K^{-}\pi^{+}$, the $D^{0}$ mass is measured to be

$\displaystyle M(D^{0})\phantom{ghd}$ $\displaystyle=$
$\displaystyle~{}1864.75\pm 0.15\,({\rm stat})\pm 0.11\,({\rm
syst})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$

The mass differences

$\displaystyle M(D^{+})-M(D^{0})$ $\displaystyle=$ $\displaystyle 4.76\pm
0.12\,({\rm stat})\pm 0.07\,({\rm
syst})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle
M(D^{+}_{s})-M(D^{+})$ $\displaystyle=$ $\displaystyle\phantom{00}98.68\pm
0.03\,({\rm stat})\pm 0.04\,({\rm
syst})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$

are measured using the $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ and
$D^{+}_{(s)}\rightarrow K^{+}K^{-}\pi^{+}$ modes.

Submitted to JHEP

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk57, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H.
Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A.
Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van
Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El
Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez
Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,
C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2,
M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, B. Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37,
T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, L. Martin54, A. Martín Sánchez7, M. Martinelli40,
D. Martinez Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A.
McNab53, R. McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40,
D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53,
P. Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren 35,o, B.K. Pal57,
A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, V.
Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K.
Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

Mesons are colourless objects composed of a quark-antiquark pair bound via the
strong interaction. Measurements of meson masses provide observables that can
be compared to theoretical predictions. For the case of $B$ mesons, precision
measurements have been reported in recent years by several experiments [1, 2,
3]. In contrast, few precision $D$ meson mass measurements exist.

For the $D^{0}$ meson111The inclusion of charge conjugate states is implied.
the current average of $M(D^{0})=1864.91\pm
0.17\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, quoted by the Review of
Particle Physics [4], is dominated by the measurements of the CLEO [5] and
KEDR [6] collaborations. Current knowledge of the masses of the $D^{+}$ and
$D^{+}_{s}$ mesons, and the mass splitting between these states, is more
limited. The most precise determination of the $D^{+}$ mass is made by the
KEDR collaboration [6] resulting in $M(D^{+})=1869.53\pm
0.49\,\mathrm{(stat)}\,\pm
0.20\,\mathrm{(syst)}\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In addition,
two measurements of the mass splitting between the $D^{+}$ and $D^{0}$ mesons
by the MRK2 [7] and LGW [8] collaborations have been reported. These are
averaged [4] to give $M(D^{+})-M(D^{0})=4.76\pm
0.28\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. No absolute measurement of
the $D^{+}_{s}$ mass with a precision better than the
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ level exists and the reported
values are not in good agreement [4]. More precise measurements of the mass
difference relative to the $D^{+}$ meson have been reported by several
collaborations [9, 10, 11, 12, 13]. These are averaged [4] to give
$M(D^{+}_{s})$ $-$ $M(D^{+})=98.85\pm
0.25\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The fit of open charm mass
data [4] leads to $M(D^{+}_{s})=1968.49\pm
0.32\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Though this value is
significantly more precise than the direct measurement, it would still
dominate the systematic uncertainty on the measurement of the $B_{c}^{+}$ mass
in the $B_{c}^{+}\rightarrow J/\psi D^{+}_{s}$ decay mode [14].

Recent interest in the $D^{0}$ mass has been driven by the observation of the
$X(3872)$ state, first measured by the Belle experiment [15] and subsequently
confirmed elsewhere [16, 17, 18, 19, 20] . This state, with $J^{PC}=1^{++}$
[21], does not fit well into the quark model picture, and exotic
interpretations have been suggested: for example that it is a tetraquark [22]
or a loosely bound deuteron-like $D^{*0}{\overline{D}}{}^{0}$ ‘molecule’ [23].
For the latter interpretation to be valid, the mass of the $X(3872)$ state
should be less than the sum of the $D^{*0}$ and ${D}^{0}$ masses. Using the
fitted value of the $D^{0}$ mass and the measured values for the other
quantities quoted in Ref. [4], the binding energy ($E_{B}$) in this
interpretation can be estimated to be

$\displaystyle E_{\textrm{B}}$ $\displaystyle=$ $\displaystyle
M(D^{0}D^{*0})-M(X(3872))$ $\displaystyle=$ $\displaystyle 2M(D^{0})+\Delta
M(D^{*0}-{D}^{0})-M(X(3872))$ $\displaystyle=$ $\displaystyle 0.16\pm
0.32{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$

Therefore, the issue of whether the $X(3872)$ can be a bound molecular state
remains open. To clarify the situation, more precise measurements of both the
$X(3872)$ and $D^{0}$ masses are needed.

In this paper, a measurement of the $D^{0}$ mass using the $D^{0}\rightarrow
K^{+}K^{-}K^{-}\pi^{+}$ decay mode is reported. This mode has a relatively low
energy release, $Q$-value, defined as the difference between the mass of the
$D$ meson and the sum of the masses of the daughter particles. Consequently,
systematic uncertainties due to the calibration of the momentum scale of the
detector are reduced. Other four-body $D^{0}$ decay modes are used to provide
a cross-check of the result. In addition, precision measurements of the
$D^{+}-{D}^{0}$ and $D^{+}_{s}-{D}^{+}$ mass differences are made. For the
mass difference measurements the $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$
mode is used, together with the $D^{+}_{(s)}\rightarrow K^{+}K^{-}\pi^{+}$
decay, since these modes have similar $Q$-values.

##  2 Detector and dataset

The analysis uses data, corresponding to an integrated luminosity of $\rm
1.0~{}fb^{-1}$, collected in $pp$ collisions at a centre-of-mass energy of
$\sqrt{s}$ = 7 $\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment during
2011\. The detector response is studied using a simulation. Proton-proton
collisions are generated using Pythia 6.4 [24] with the configuration
described in Ref [25]. Particle decays are then simulated by EvtGen [26] in
which final state radiation is generated using Photos [27]. The interaction of
the generated particles with the detector and its response are implemented
using the Geant4 toolkit [28, *Agostinelli:2002hh] with the settings described
in Ref. [30].

The LHCb detector [31] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$. It includes a high precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$ interaction
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\rm\,Tm}$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream. The polarity
of the dipole magnet is reversed at intervals that correspond to roughly
$0.1\,\mbox{\,fb}^{-1}$ of collected data in order to minimize systematic
uncertainties. The combined tracking system has momentum resolution $\Delta
p/p$ that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6%
at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger [32] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction. Samples of open charm mesons produced directly in the
primary $pp$ interaction (refered to as ‘prompt’) and in semileptonic decays
of $b$-hadrons are selected by the trigger. Though the prompt sample is larger
in size, cuts on the decay time of the $D$ meson are applied at the trigger
level to reduce the output rate. As the reconstructed mass and decay time are
correlated, these cuts bias the mass measurement. In contrast, no cuts on the
$D$ decay time are applied at the trigger level for the semileptonic sample,
which is therefore used for this analysis.

The measurements require the momenta of the final state particles to be
determined accurately. The procedure used to calibrate the momentum scale of
the tracking system for this study is discussed in detail in Ref. [33]. It is
based upon large calibration samples of
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
collected concurrently with the dataset used for this analysis. The use of the
large $J/\psi$ dataset allows to correct for variations of the momentum scale
at the level of $10^{-4}$ or less that occur over the course of the data-
taking period whilst the use of the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ allows the
momentum scale to be determined as a function of the $K^{+}$ kinematics. The
accuracy of the procedure has been checked using other fully reconstructed $B$
decays together with two-body $\Upsilon(nS)$ and $K^{0}_{\rm\scriptscriptstyle
S}$ decays. In each case the deviation of the measured mass from the expected
value is converted to an estimate of the bias on the momentum scale ($\alpha$)
taking into account relativistic kinematics and QED radiative corrections. The
largest value of $|\alpha|$ found in these studies is $0.03\,\%$ for the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay mode.
Conservatively, this is taken as the uncertainty on the calibrated momentum
scale. This leads to the largest contribution to the systematic uncertainty on
the mass measurements.

##  3 Selection

The selection uses only well reconstructed charged particles that traverse the
entire tracking system. All charged particles are required to be within the
angular acceptance of the spectrometer. This corresponds to 300 mrad in the
bending plane of the dipole magnet and 250 mrad in the orthogonal plane. In
addition, the final state particles are required to have $p_{\rm T}$ greater
than 300 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Further background
suppression is achieved by exploiting the fact that the products of heavy
flavour decays have a large distance of closest approach (‘impact parameter’)
with respect to the $pp$ interaction vertex in which they were produced. The
impact parameter $\chi^{2}$ with respect to any primary vertex is required to
be larger than nine. Fake tracks created by the reconstruction are suppressed
by cutting on the output of a neural network trained to discriminate between
these and real particles. This cut also removes candidates where one of the
charged hadrons has decayed in flight. To select well-identified kaons (pions)
the difference in the logarithms of the global likelihood of the kaon (pion)
hypothesis relative to the pion (kaon) hypothesis provided by the ring-imaging
Cherenkov detectors is required to be greater than five (zero).

Charged particles selected in this way are combined to form $D^{0}\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$, $D^{0}\rightarrow K^{+}K^{-}K^{-}\pi^{+}$ and
$D^{+}_{(s)}\rightarrow K^{+}K^{-}\pi^{+}$ candidates. To eliminate kinematic
reflections due to misidentified pions, the invariant mass of at least one
kaon pair is required to be within $\pm
12~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal value of the
$\phi$ meson mass [4]. This requirement means that the $D$ meson sample is
dominated by decays containing an intermediate $\phi$ meson. A fit requiring
the final state particles to originate from a common point is made and the
$\chi^{2}$ per degree of freedom ($\chi^{2}/$ndf) of this fit is required to
be less than five. In order to remove poorly reconstructed candidates, a cut
is made on the uncertainty of the reconstructed invariant mass estimated by
propagation of the individual track covariance matrices. The value of this cut
depends on the decay mode under consideration and is chosen such that the bulk
of the distribution is kept and only events in the tail are rejected. In a few
percent of the events the reconstruction procedure gives rise to duplicate
candidates. Therefore, if two or more candidates that are separated by less
than 0.05 in pseudorapidity and $50~{}$mrad in azimuthal angle are found
within one event, only that with the best $D$ vertex $\chi^{2}$ is kept.

Each candidate $D$ meson, selected in this way, is combined with a well-
identified muon that is displaced from the $pp$ interaction vertex (impact
parameter $\chi^{2}>4$) and that has $p_{\rm T}$ larger than
$800~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ to form a $B$ candidate. A fit is
made requiring the muon and the $D$ candidate to originate from a common point
and the $\chi^{2}$ per degree of freedom of this fit is required to be less
than five. To select semileptonic $B$ decays, the invariant mass of the $B$
candidate is required to be in the range
$2.5-6.0~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In principle, the large
combinatorial background can be further reduced by cutting on the decay time
of the $D$ meson, but due to the correlation between the decay time and the
mass, this cut would bias the mass distribution. Therefore, a cut requiring
significant displacement between the $b$-hadron decay vertex and the
associated $pp$ interaction vertex is applied. This achieves high signal
purity whilst not biasing the distribution of the $D$ decay time.

##  4 Fit Results

The $D$ meson masses are determined by performing extended unbinned maximum
likelihood fits to the invariant mass distributions. In these fits the
background is modelled by an exponential function and the signal by the sum of
a Crystal Ball [34] and a Gaussian function. The Crystal Ball component
accounts for the presence of the QED radiative tail. Alternative models for
both the signal and background components are considered as part of the
studies of the systematic uncertainties. The model for the signal shape
contains six parameters:

  * •

a common mean value for the Gaussian and Crystal Ball components;

  * •

the widths of the Gaussian ($\sigma_{\mathrm{G}}$) and the Crystal Ball
($\sigma_{\mathrm{CB}}$) components;

  * •

the transition point ($a$) and exponent ($n$) of the Crystal Ball component;

  * •

the relative fraction of the Crystal Ball ($f_{\textrm{CB}}$) component.

To reduce the number of free parameters in the fit, $a$, $n$ and
$f_{\mathrm{CB}}$ together with the ratio of $\sigma_{\mathrm{CB}}$ to
$\sigma_{\mathrm{G}}$, are fixed using a simulation that has been tuned to
reproduce the mass resolution observed in data for the $B^{+}\rightarrow
J/\psi K^{+}$ and $B^{+}\rightarrow J/\psi K^{+}\pi^{-}\pi^{+}$ decay modes.
By fixing the ratio of $\sigma_{\mathrm{CB}}$ to $\sigma_{\mathrm{G}}$ the
resolution model is constrained up to an overall resolution scale factor that
is close to unity.

The Crystal Ball function describes the effect of the radiative tail far from
the peak well. However, close to the peak its shape is still Gaussian, which
results in a bias on the fitted mass that scales with the $Q$-value of the
decay mode. This effect is studied using Photos [27] to model the effect of
QED radiative corrections. The size of the bias is found to be $0.03\pm
0.01\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the $D^{0}\rightarrow
K^{+}K^{-}K^{-}\pi^{+}$ mode. For the $D^{0}\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$, $D^{+}\rightarrow K^{+}K^{-}\pi^{+}$ and
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decay modes a value of $0.06\pm
0.01~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is found. These values are
used to correct the mass measurements. The effect cancels in the measurement
of the mass differences.

The resulting fits for the $D^{0}$ decay modes are shown in Fig. 1 and that
for the $K^{+}K^{-}\pi^{+}$ final state in Fig. 2. The values obtained in
these fits are summarized in Table 1. The resulting values of the $D^{+}$ and
$D^{+}_{s}$ masses are in agreement with the current world averages. These
modes have relatively large $Q$-values and consequently the systematic
uncertainty due to the knowledge of the momentum scale is at the level of
$0.3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Hence, it is chosen not to
quote these values as measurements. Similarly, the systematic uncertainty due
to the momentum scale for the $D^{0}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ mode
is estimated to be $0.2~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the
measured mass in this mode is not used in the $D^{0}$ mass determination.

Figure 1: Invariant mass distributions for the (a) $K^{+}K^{-}\pi^{+}\pi^{-}$
and (b) $K^{+}K^{-}K^{-}\pi^{+}$ final states. In each case the result of the
fit described in the text is superimposed (solid line) together with the
background component (dotted line). The pull, i.e. the difference between the
fitted and measured value divided by the uncertainty on the measured value, is
shown below each plot.

Figure 2: Invariant mass distribution for the $K^{+}K^{-}\pi^{+}$ final state.
The result of the fit described in the text is superimposed (solid line)
together with the background component (dotted line). The pull, i.e. the
difference between the fitted value and the measured value divided by the
uncertainty, is shown below the plot. Table 1: Signal yields, mass values,
resolution scale factors and binned $\chi^{2}$/ndf (using 100 bins) obtained
from the fits shown in Fig. 1 and Fig. 2 together with the values corrected
for the effect of QED radiative corrections as described in the text.

|  | Fitted mass | Corrected mass | Resolution |
---|---|---|---|---|---
Decay mode | Yield | [$\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$] | [$\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$] | scale factor | $\chi^{2}$/ndf
$D^{0}\,\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ | $\phantom{111}4608\pm 89\phantom{1}$ | $1864.68\pm 0.12$ | $1864.74\pm 0.12$ | $1.031\pm 0.021$ | 0.83
$D^{0}\,\rightarrow K^{+}K^{-}K^{-}\pi^{+}$ | $\phantom{1111}849\pm 36\phantom{1}$ | $1864.73\pm 0.15$ | $1864.75\pm 0.15$ | $0.981\pm 0.042$ | 0.92
$D^{+}\rightarrow K^{+}K^{-}\pi^{+}$ | $\phantom{1}68,787\pm 321$ | $1869.44\pm 0.03$ | $1869.50\pm 0.03$ | $0.972\pm 0.003$ |
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ | $248,694\pm 540$ | $1968.13\pm 0.03$ | $1968.19\pm 0.03$ | $0.971\pm 0.002$ | 2.5

The quality of the fits is judged from the $\chi^{2}$/ndf, quoted in Table 1,
and the fit residuals. It has been checked using simulated pseudo-experiments
that the sizeable trends seen in the residuals for the $K^{+}K^{-}\pi^{+}$
mode, where the dataset is largest, do not bias the mass difference
measurement. The fitted resolution scale factors are all within a few percent
of unity, indicating that the calibration parameters obtained from the $B^{+}$
study are applicable in this analysis. The uncertainties on the masses
reported by the fits are in good agreement with the results obtained in
pseudo-experiments.

Using the values in Table 1, the mass differences are evaluated to be

$M(D^{+})-M(D^{0})$ $~{}=~{}$ | 4.76 $\,\,\pm\,$ | 0.12 (stat) MeV/$c^{2}$ | ,
---|---|---|---
$M(D^{+}_{s})-M(D^{+})$ $~{}=~{}$ | 98.68 $\,\,\pm\,$ | 0.03 (stat) MeV/$c^{2}$ | ,

where the uncertainties are statistical only.

##  5 Systematic uncertainties

To evaluate the systematic uncertainty, the complete analysis is repeated,
including the track fit and the momentum scale calibration when needed,
varying within their uncertainties the parameters to which the mass
determination is sensitive. The observed changes in the central values of the
fitted masses relative to the nominal results are assigned as systematic
uncertainties.

The dominant source of uncertainty is the limited knowledge of the momentum
scale. The mass fits are repeated with the momentum scale varied by $\pm
0.03\,\%$. A further uncertainty is related to the understanding of the energy
loss in the material of the tracking system. The amount of material traversed
in the tracking system by a particle is known to 10 % accuracy [35].
Therefore, the magnitude of the energy loss correction in the reconstruction
is varied by $\pm 10\,\%$.

Other uncertainties arise from the fit model. To evaluate the impact of the
signal model, a fit is performed where all signal parameters are fixed
according to the values found in the simulation and a second fit where the
parameters $\sigma_{\mathrm{G}}$ and $\sigma_{\mathrm{CB}}$ are allowed to
vary while keeping the relative fraction, $f_{\mathrm{CB}}$, of the two
components fixed. The larger of the differences to the default fit result is
assigned as an estimate of the systematic uncertainty. Similarly the effect of
the background modelling is estimated by replacing the exponential function
with a first-order Chebychev polynomial. The shifts of the mass values
observed in these tests are generally much smaller than
$0.01\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and are assigned as
systematic uncertainties. For the $K^{+}K^{-}\pi^{+}$ fit further cancellation
occurs in the mass difference. It is concluded that the details of the fit
model have little effect on the presented measurements.

An additional uncertainty arises from the knowledge of the value of the
$K^{+}$ mass, $m_{K^{\pm}}=493.677\pm
0.016{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [4]. The effect of this
uncertainty on the measurements has been evaluated from simulation studies.

The systematic uncertainties on the measured masses and mass differences are
summarized in Table 2. The uncertainties related to the momentum scale and
energy loss correction are fully correlated between the measurements.

Table 2: Systematic uncertainties (in ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$) on the mass measurements and on their differences. Source of uncertainty | $M(D^{0})$ | $M(D^{+})-M(D^{0})$ | $M(D^{+}_{s})-M(D^{+})$
---|---|---|---
Momentum scale |  $<$0.09 |  $<$0.04 |  $<$0.04
Energy loss correction |  $<$0.03 |  $<$0.06 |  $<$0.01
$K^{\pm}$ mass |  $<$0.05 |  $<$0.01 |  $<$0.01
Signal model |  $<$0.02 |  $<$0.01 |  $<$0.01
Background model |  $<$0.01 |  $<$0.01 |  $<$0.01
Quadratic sum |  $<$0.11 |  $<$0.07 |  $<$0.04

Various cross-checks of the measurements are made. Two checks are related to
the knowledge of the tracking system alignment. First, a study has been
performed where particle trajectories are reconstructed without using the
information related to the tracking detector located before the entrance of
the spectrometer magnet. This information is not required to form a track but
improves the momentum resolution by $10-20\,\%$. The second test is to vary
the track slopes in the vertex detector by the uncertainty of $2\times
10^{-4}$ on the length scale of the detector described in Ref. [36]. The
results obtained in these studies are consistent with those presented here and
no additional uncertainty is assigned.

A further check for the $D^{0}$ mass measurement is the comparison of the
measured mass in the $D^{0}\rightarrow K^{+}K^{-}K^{-}\pi^{+}$ mode with that
obtained in the three other four-body modes. Systematic effects related to the
momentum scale will affect modes with a high $Q$-value more than those with
low $Q$-values. The relationship between the reconstructed mass ($m$) and the
momentum scale ($\alpha$) after a first-order Taylor expansion in
$m^{2}/p^{2}$ is

$m^{2}=\frac{m^{2}_{\mathrm{true}}-f}{(1-\alpha)^{2}}+f,$ (1)

where

$f=p\sum\frac{m_{i}^{2}}{p_{i}},$ (2)

$p$ is the total momentum of the decaying meson and $p_{i}$ and $m_{i}$ are
the momenta and masses of the daughter particles. This formalism assumes that
there are no additional differences affecting the momentum scale between the
modes such as differences in track kinematics or the effect of QED radiative
corrections. For each decay mode the average value of $f$ is obtained from the
data using the sPlot technique [37] with the mass as the control variable to
subtract the effect of background. The values obtained in this way are in good
agreement with those found in the simulation. In Fig. 3 the measured $D^{0}$
mass is plotted versus $f$ for the four-body decay modes studied here. The
shaded area on this plot corresponds to the assigned systematic uncertainty of
$0.03\,\%$ on the momentum scale. Though there is evidence of a systematic
effect for the low $f$-value modes it is accounted for by the assigned
uncertainty.

Figure 3: Measured $D^{0}$ mass versus $f$ as defined in Eq. 2. The (yellow)
shaded area corresponds to a systematic uncertainty on the momentum scale of
$0.03\,\%$ centred on the result for the $D^{0}\rightarrow
K^{+}K^{-}K^{-}\pi^{+}$ mode (horizontal dashed line). Only the
$D^{0}\rightarrow K^{+}K^{-}K^{-}\pi^{+}$ mode, where the systematic
uncertainty is lowest, is used to determine the $D^{0}$ mass.

The dataset has also been divided according to the magnet polarity and data-
taking period and for the charged modes by the sign of the product of the
magnet polarity and the $D$ meson charge. In addition, for modes where the
event samples are sizable the measurements are repeated in bins of the $D$
meson kinematic variables. None of these tests reveal any evidence of a
systematic bias.

##  6 Summary

Measurements of $D$ meson masses and mass differences have been performed
using $pp$ collision data, corresponding to an integrated luminosity of
$1.0~{}\mbox{\,fb}^{-1}$ collected at a centre-of-mass energy of
$\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ with the LHCb detector. The
results are

$M(D^{0})$ $~{}=~{}$ | 1864.75 $\,\,\pm\,$ | 0.15 (stat) $\pm\,$ | 0.11 (syst) MeV/$c^{2}$ | ,
---|---|---|---|---
$M(D^{+})$ $-$ $M(D^{0})$ $~{}=~{}$ |  1114.76 $\,\,\pm\,$ | 0.12 (stat) $\pm\,$ | 0.07 (syst) MeV/$c^{2}$ | ,
$M(D^{+}_{s})$ $-$ $M(D^{+})$ $~{}=~{}$ | 98.68 $\,\,\pm\,$ | 0.03 (stat) $\pm\,$ | 0.04 (syst) MeV/$c^{2}$ | .

The dominant systematic uncertainty is related to the knowledge of the
momentum scale.

As shown in Table 3, these measurements are in agreement with previous
measurements. The results for the mass differences have smaller uncertainty
than any previously reported value. The measured value of the $D^{0}$ mass has
a similar precision to the published CLEO result [5]. Including this result in
the determination of the $X(3872)$ binding energy given in Section 1 gives
$E_{B}=0.09\pm 0.28{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This reinforces
the conclusion that if the $X(3872)$ state is a molecule it is extremely
loosely bound.

Table 3: LHCb measurements, compared to the best previous measurements and to the results of a global fit to available open charm mass data. The quoted uncertainties are the quadratic sums of the statistical and systematic contributions. All values are in ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. | LHCb | Best previous |
---|---|---|---
Quantity | measurement | measurement | PDG fit [4]
$M(D^{0})$ | $1864.75\pm 0.19$ |  $1864.85\pm 0.18$ [5] | $1864.86\pm 0.13$
$M(D^{+})-M(D^{0})$ | $\phantom{111}4.76\pm 0.14$ |  $\phantom{111}4.7\phantom{0}\pm 0.3\phantom{0}$ [7] | $~{}~{}~{}~{}~{}4.76\pm 0.10$
$M(D^{+}_{s})-M(D^{+})$ | $\phantom{11}98.68\pm 0.05$ |  $\phantom{111}98.4\phantom{0}\pm 0.3\phantom{0}$ [10] | $~{}~{}~{}98.88\pm 0.25$

The measurements presented here, together with those given in Ref. [4] for the
$D^{+}$ and $D^{0}$ mass, and the mass differences $M(D^{+})-M(D^{0})$,
$M(D^{+}_{s})-M(D^{+})$ can be used to determine a more precise value of the
$D^{+}_{s}$ mass

$M(D^{+}_{s})=1968.19\pm 0.20\pm 0.14\pm
0.08{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$

where the first uncertainty is the quadratic sum of the statistical and
uncorrelated systematic uncertainty, the second is due to the momentum scale
and the third due to the energy loss. This value is consistent with, but more
precise than, that obtained from the fit to open charm mass data,
$M(D^{+}_{s})=1968.49\pm 0.32~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [4].

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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