# Do Neural Networks Generalize from Self-Averaging Sub-classifiers in the
Same Way As Adaptive Boosting?

Michael Sun  Stanford University, Stanford, CA  Peter Chatain  Stanford
University, Stanford, CA

(March 2022)

###### Abstract

In recent years, neural networks (NN’s) have made giant leaps in a wide
variety of domains. NN’s are often referred to as “black box” algorithms due
to how little we can explain their empirical success. Our foundational
research seeks to explain why neural networks generalize. A recent advancement
derived a mutual information measure for explaining the performance of deep
NN’s through a sequence of increasingly complex functions. We show deep NN’s
learn a series of boosted classifiers whose generalization is popularly
attributed to self-averaging over an increasing number of interpolating sub-
classifiers. To our knowledge, we are the first authors to establish the
connection between generalization in boosted classifiers and generalization in
deep NN’s. Our experimental evidence and theoretical analysis suggest NNs
trained with dropout exhibit similar self-averaging behavior over
interpolating sub-classifiers as cited in popular explanations for the post-
interpolation generalization phenomenon in boosting.

##  1 Introduction

###  1.1 Introducing and Motivating the problem

[6] showed neural networks are universal function approximators that can
achieve remarkably low bias on estimating highly complex functions on high
dimensional data, producing bountiful successes in a wide range of domains and
applications. This makes it all the more surprising, in the view of
statistical learning theory, why they can generalize well to unseen data.
There have been considerable advances in demonstrating the generalization of
deep neural networks. Characterizing when NNs will generalize in terms of its
capacity has informed many common heuristics and practices such as dropout in
the machine learning community, and has driven the rapid adoption of them in
real-world applications. However, explaining why they generalize still remains
an open problem.

Of particular interest is using simpler models to characterize and decompose
what a NN is learning through looking at their mutual information. Boosting is
an ensemble method to make a stronger classifier out of weaker classifiers,
with each classifier is explainable in their features and interpretable in
their decisions. In the popular variant adaptive boosting, each weak
classifier is trained in succession, and the next weak classifier trains on
re-weighted data which focuses on examples the ensemble does poorly on.
Adaptive boosting is explainable as a weighted majority vote algorithm
(described by its authors [5]), where each vote comes from a weak learner that
has an easily explained decision rule. Moreover, a familiar “self-averaging”
view from the original authors on Dropout has recently gained a lot of
consensus for explaining why Adaboost rarely overfit in practice [7] [18].
Within an ensemble of classifiers, self-averaging is the process which
stabilizes the fit to regions with signal, while continuing to localize the
effect of noisy points on the overall fit. Similarly, dropout is often used in
practice to reduce the NN from overfitting. Dropout randomly selects a subset
of the neurons to ignore while training, and it selects a different subset at
every epoch. When making predictions, all of the neurons are used. We are
inspired by dropout to view a NN as an ensemble of sub-classifiers, and relate
each iteration of boosting to some sub-network.

The time is ripe for us to bridge these two parallel developments on
generalization amongst empirical risk minimizers. This connection can go a
long way in uniting two traditionally separate fields of research – boosting
and deep learning – and help ground the success of neural networks in more
established territory.

###  1.2 Your contributions

Previous work has conjectured that neural networks learn a series of
increasingly complex functions ([11]), although the sequence and notion of
complexity are not defined. We will define the series of increasingly complex
functions as adaptive boosted classifiers with well-characterized growth in
its VC dimension in terms of the number of boosting rounds. Each complexity
rung from this series corresponds to a distinct phase of training.

To justify this definition, we inductively show a similarity between the
learning outcomes in a single phase between a single hidden layer neural
network and boosted classifier. Namely, the NN’s mutual information with $Y$,
the label distribution, can be explained away by a boosted algorithm from the
next phase, and the boosted algorithm’s mutual information with $Y$ can be
explained away by the NN in the next phase. In addition, we run extensive
experiments demonstrating that NNs learn and retain increasingly complex
hypotheses from adaptive boosting. In the first phase of training, the mutual
performance between a single weak learner and the NN is perfect. Then, as the
NN learns increasingly complex concepts, this simple concept is not forgotten.
We show that this same pattern holds in further phases of training for boosted
algorithms with more weak learners. We corroborate these claims with
qualitative evidence by investigating the learning order of examples for each
model and find similar preference of example difficulty within the same
corresponding phase.

We further investigate the connection between boosting and neural networks by
studying a chosen ensemble of sub-networks chosen as in dropout as shown by
[15] and [1]. In boosting, each weak learner is encouraged to perform well on
different subsets of the data which decreases the correlation between each of
them. We prove that training only sub-networks within our chosen ensemble
similarly decreases the correlation between any two hidden neurons in a
simplified setting. We also experimentally validate the self averaging
behavior of NNs trained with our modified version of dropout by viewing each
sub-network created by dropout as an independent learner. When compared with
neural networks that do not use dropout, the self averaging effect is more
pronounced.

###  1.3 Related Research

There have been many attempts to formulate and motivate the search for simpler
neural networks. For a short early history of the development of such
techniques and how they encourage simplicity, one is encouraged to read [13].
Among these, of particular interest to us is dropout, a technique for
addressing overfitting by averaging over an exponential number of “thinned”
networks [15]. This is elaborated by [1]’s recursive view of the averaging
properties of dropout.

However, such techniques that seek to design simpler neural networks do not
yield simple explanations for why the resulting models generalize.

As another attempt, there have been many attempts to explain generalization
seen in broader classes of interpolating classifiers. This is the case with
random forests, adaptive boosting [18], and nearest-neighbor schemes [3].
Meanwhile, understanding of generalization in interpolating neural networks
only began advancing recently, with theoretical analysis in more realistic
settings with e.g. noisy labels and nonlinear learning dynamics by [4] still
in its nascent stages.

We seek to bridge these two complementary fields of research by showing deep
NNs learn a series of boosted classifiers, whose generalization is popularly
attributed to self-averaging over an increasing number of interpolating sub-
classifiers. Towards that end, our survey of what’s known will overview the
literature on neural network complexity and adaptive boosting separately.
Then, we will address previous attempts to explain neural networks by simpler
models, identify gaps in understanding, then argue why the novel connection we
establish between adaptive boosting and neural networks can fill in some of
those gaps.

##  2 Scientific Background

How do different choices of low complexity models help us understand and
explain the generalization of a neural network?  Do neural networks, under
certain hyperparameters, generalize through a self averaging effect as in
boosting?

###  2.1 Motivation

In recent years, neural networks have made giant leaps in a wide variety of
domains. Neural networks are often referred to as “black box” algorithms due
to how little we can explain its empirical success. A lack of understanding
raises concerns including but not limited to explainability, interpretability,
adversarial robustness, and bias.

Our foundational research seeks to understand how and why neural networks
generalize by using simpler, less complex models. Aristotle described one of
the earliest forms of what we now call Occam’s Razor in his Posterior
Analytics by saying “We may assume the superiority … [all else being equal] of
the demonstration which derives from fewer postulates or hypotheses.”
(Aristotle, Posterior Analytics, p. 150). From this theoretical framework,
neural networks with lower complexity are preferred. In a practical setting,
simple explanations for what a neural network learned and how it did so can
bolster confidence for deployment as well as bring to light pitfalls.

###  2.2 What’s Known

Developing a framework for understanding when and why neural networks work has
been a longstanding problem. Early studies by [10] found that NNs learn
increasingly good representations in deeper layers, where good is defined as
how much predictive signal we can extract (using PCA) from the associated
Kernel matrix using a linear classifier. This understanding has proved
valuable for tasks such as transfer learning. On the other hand, they also
found that Neural networks trained with gradient descent struggle generalizing
on few-shot “reasoning” tasks [13]. An example of such a task is predicting
the number of on-bits in a 100-dimensional binary vector from just 3 examples.
ML research on generalization today generally adopts gradient descent, instead
studying the statistical assumptions/interpretations behind techniques of
constraining the neural network. Common notions of “complexity” are in terms
of the number and magnitude of parameters.

One measure for complexity is just to measure the euclidean distance between a
vector of all the weights of a model and a vector of all ones. This relates to
the more classical measures like Kolmogorov complexity, where a neural network
with most weights equal to zero have lower complexity due to fewer bits needed
to store them. A paper by [12] investigated an improvement on this by defining
a topologically derived neural persistence measure based on the idea that some
neurons contribute more to the final prediction. This work showed that finding
the right measures of complexity can explain best practices in machine
learning. In their experiments, dropout, a widely used technique for
regularization, increased the neural persistence by a statistically
significant amount.

Another measure of complexity that is not specific to neural networks is the
VC dimension, a measure of the capacity of a set of functions that can be
learned. This general measure is well-studied for many function classes that
we consider as candidates for “simple models”, from boosting classifiers to
neural networks. For boosting, we know $d_{k}<2kd\log(ke)$ where $d$ is the VC
dimension of the base hypothesis class, and $d_{k}$ is that of the boosted
hypothesis class after $k$ rounds of boosting. Moreover, the authors of
Adaboost [5] gave various bounds on the generalization error in terms of the
training error and $d_{k}$. Bounds on the VC dimension of MLPs have also been
given for various choices of activations by [14]. For example, with sigmoid
activation, the VC dimension of a MLP with neurons V and connections E is
$\Theta(|E|^{2})$ and $O(|E|^{2}|V|^{2})$. Thus, the VC dimension has the
advantage of being general to any function class, and will be part of one of
our conjectures relating boosted classifiers and neural networks.

Adaptive boosting was introduced by [5] as an improved boosting algorithm
which adjusts adaptively to the errors of weak hypotheses found in previous
iterations. Boosting has been observed to rarely overfit in practice, and many
recent explanations have been proposed to explain why. These include margin-
based explanations by [2] in terms of “mirror descent applied to the problem
of maximizing the smallest margin in the training set under suitable
separability conditions,” and they include optimization explanations in terms
of the minimization of a convex loss function or “a procedure to search
through the space of convex combinations of weak learners or base
classifiers.” Notably, [18] and [9] suggested that Adaboost exhibits the self-
averaging mechanism of many interpolating classifiers and thus generalizes for
the same reason random forests do.

More recently, it was shown by [11] that NNs trained with stochastic gradient
descent (SGD) share high performance correlation with linear classifiers in
the initial phase of training and retain it afterwards. Contemporary work by
[8] showed that the degree to which this is true depends on the learning rate.
With a small learning rate the model learns a complex, non-linear decision
boundary because it will memorize complex patterns wherever possible. They
demonstrated this qualitatively by adding a memorizable patch to CIFAR-10
images, and showing the small learning rate model maximizes patch accuracy
from the start. Their claims rely on the fact neural networks have a non-
convex loss landscape, and may not hold outside the setting of SGD and a
sufficiently large learning rate.

###  2.3 What’s Still Missing

As described in [11]’s paper, there are gaping holes in their central
conjecture. Namely, there exists a “sequence” of “increasingly complex”
functions $g_{i}$ and timesteps $T_{i}$ for which up until $T_{i}$, the neural
network F learns nothing beyond $g_{i}$ and after $T_{i}$, retains it. Neither
the function sequence or notion of complexity is defined. The combination of
boosting and VC dimension have not been investigated as candidates to define
this conjecture.

More broadly, the machine learning community can’t always predict in
understandable terms when a model will or will not generalize. Measuring the
performance correlation of two models is a new line of inquiry in this space,
and has yet to be explored with other architectures than linear models.

To address the stochasticity of training, we provide a theoretically motivated
training setup which corroborates our claims in two ways: 1) a high
performance correlation between neural networks and boosted classifiers and 2)
a similar distribution of errors over examples. We also formalize a general
connection between the self-averaging property in dropout and boosted
classifiers, and include in our analysis the degrees of sensitivity of this
conjecture to various choices of hyperparameters.

##  3 Suggested Research

###  3.1 Meta-goals

Explaining the generalization of all neural networks by a simple model class
may be intractable (otherwise, there’d be no need for neural networks).
Instead, we think different choices of “low complexity models” can better
explain different constrained subclasses of neural networks. We want to
identify one such model class and the corresponding subclass of neural
networks. More formally, we want to explain what a NN is learning at each
phase of training corresponding to training iterations $T_{i},T_{i+1}$ for
$i>=0$. Denote $F_{i}$ as the neural network at $T_{i}$. We want to find
functions $G_{i}:i>=0$ of increasing “complexity” such that
$I(F_{i};Y|G_{i+1})\simeq 0$, and $I(G_{i};Y|F_{i+1})\simeq 0$ for all $i$. To
fill in the missing parts, we want to materialize $G_{i}:i>=0$, and try
suitable definitions of “complexity”. A stretch goal is to also characterize
the timesteps $T_{i}$, so we know when we can expect neural networks to
generalize. A third meta-goal which we describe in detail later is to
“explain” what $F$ is learning. We can do so if we can see if $F$ learns
increasingly “difficult” examples in the same order as $G_{i}$.

###  3.2 Specific Approaches

Boosting Under VC dimension bounds given on Adaboost by [2] and [5], we can
formalize the complexity of the weak learners by varying the number of
boosting iterations. Towards achieving the meta-goal about a series of
increasingly complex learners, we can simultaneously train a neural network
and Adaboost, checkpointing (saving) the neural network periodically. Here,
the series of complex learners are boosted algorithms with more learners.
Similarly, we checkpoint the final boosted classifier after each boosting
round. Afterwards, we can plot the relevant quantities $I(F;Y|G)$ and
$I(G;Y|F)$ for all pairwise combinations $(F,G)$ over the checkpoints. We
expect to see a similar phase separation as in [11]’s paper.

Learning Order It would go a long way to “explaining” what $F$ is learning if
we can see if it’s learning increasingly “difficult” examples in the same
order as $G_{i}$. To do so, we can define the difficulty of an example as the
difficulty of the class it resides in, and its euclidean distance of its
embedding from the mean of embeddings in that class. Then, we can measure
progressive error on examples of various difficulty levels, and compare that
between the neural network and boosted classifiers. In particular, we can take
the embedding of examples, obtained using the pre-final layer. This can lead
to a qualitative measure to explain similarity in how the model learns. The
difficulties of the classes for common benchmark datasets have been explored
in other works. As shown by [19], different classes in CIFAR-10 are more or
less difficult. As shown by [17], “discontinuous” phase transitions are
observed by sweeping the Beta parameter of a binary classification loss
function, which controls between compression (simplicity) and accuracy. These
phase transitions correspond to new classes of examples the model learns.
Plotting the accuracy against Beta shows many discontinuities, at which the
model’s accuracy suddenly “jumps” upon learning either to recognize a new
class, or to discriminate between samples of two visually similar classes.
Thus, we may use the order of classes as a proxy for difficulty. Afterwards,
we can also see if the times at which these phase transitions occur for a
training neural network map onto the times at which the neural network
correlates with increasingly complex functions $G_{i}$ , as will be defined in
Theorem 1.

###  3.3 Specific Questions

We plan to directly extend [11]’s conjecture. We will replace g with a
recursive definition of a class of functions, and fill in the notion of
complexity as the VC dimension while keeping the same mutual performance
correlation.

###### Theorem 1

Let $G_{i}=\sum_{j=1}^{i}h_{j}$, where all $h_{j}\in H$ (hypothesis class),
and each $h_{j}$ is the (empirical risk minimizer) ERM on $S_{i}\sim P_{i},D$.
Let $F$ be $\\{f:f(x)=\sum_{j=1}^{k}v_{j}f_{j}(x),\text{ and
}f_{j}(x)=ReLU(\sum_{i=1}^{d}w_{i,j}x_{i})\\}$ denoting the set of one-hidden-
layer neural networks with k hidden units and ReLU activations. Denote $F_{i}$
as the neural network at $T_{i}$. Then

$I(F_{i};Y|G_{(}i+1))\simeq 0$ (1)

and

$I(G_{i};Y|F_{(}i+1))\simeq 0$ (2)

for all $i$ from $1$ to $J$, and is statistically significant compared to
increasingly better random classifiers $[R_{i}:1<=i<=J]$, that satisfy
$I(R_{i};Y)=I(F_{i};Y)$.

Note that $S_{i}\sim P_{i},D$ corresponds to the sample reweighting scheme in
Adaboost, formalized drawing training samples from a different (but not
independent) probability distribution each round. This is a reformulation of
[11]’s conjecture that is more direct: $F_{i}$ will learn $Y$ in the same way
as $G_{i}$ then retain it afterwards as it learns the more complex $G_{i+1}$.
That is equivalent to stating $F_{i}$ will never be more useful than $G_{i+1}$
in explaining $Y$ for all $i$. Vice-versa, $G_{i}$ will never be more useful
than $F_{i+1}$ in explaining $Y$, stating $F$ has learned and fully retained
$G_{i}$ .

To establish this relation, we have to draw a connection between the
iterations of boosting in adaptive boosting, and sub-networks from the neural
network. We do establish this relation in theorem 2.

###### Theorem 2

Let $G_{i}=\sum_{j}^{i}h_{j}$, where all $h_{j}\in H$ (hypothesis class), and
each $h_{j}$ minimizes risk on $S_{i}\sim P_{i},D$. Let
$f(x)=\sum_{j}^{k}v_{j}f_{j}(x)$, where
$f_{j}(x)=\text{ReLU}(\sum_{i=1}^{d}w_{i,j}x_{i})$ denote a one-hidden-layer
neural network $f$ with $k$ hidden units and ReLU activations. Indexed $f,h$
are random variables. SGD minibatch gradient descent will progressively
increase and decrease

$\text{corr}(f_{i1},h_{j1})\text{and corr}(f_{i1},f_{i2})$

under some matching $\\{i1,i2,…\\}\text{ to }\\{j1,j2,…\\}$, respectively.
Next, we define $J\subset\\{1,...,k\\}$, and $f^{J}(x)=\sum_{j\in
J}v_{j}f_{j}(x)$ as a “sub-network” of $f$. SGD gradient descent will
progressively decrease

$corr(f^{J}_{i1}(x),f^{J}_{i2}(x))$

.

Note that this theorem, if true, will mean each $f_{j}$ comes up with an
independent “hypothesis” which combines via a self-averaging mechanism to form
the final hypothesis. Implicitly, this means each neuron $f_{j}$ will have to
focus on different feature(s). Like in random forests and Adaboost, this self-
averaging mechanism of relatively independent base hypotheses can explain why
neural networks, under the common practice of dropout, can generalize well.

As pointed out by [8]’s paper, the dataset noise/signal matters. A specific
question we also want to ask is if the mutual information that is shared comes
from a specific portion of the data. To do so, we can experimentally separate
the data in, say, CIFAR-100, into easy and difficult tasks as described in the
learning order section. Then, we can plot the mutual information on each of
these subsets of data and find ones that correlate.

To make these sub-networks useful for practitioners trying to explain a NN
with boosted classifiers, we make the following conjecture.

###### Conjecture 1

Let $F(x)$ denote a one-hidden-layer neural network $f$ with $k$ hidden units
and ReLU activations. There exists some mapping $\pi:\mathds{N}\to\mathds{N}$
such that we can find $G$ of VC dimension $\pi(\text{VCDim}(F))$ for which
theorem 1 and 2 hold in a variety of settings.

In boosting, one needs to choose both the base hypothesis class and the number
of boosting rounds. With this conjecture, we envision general practices to
choose the base hypothesis class. This will substantially accelerate the
practitioner’s’ search for finding a boosted classifier which can pair with
the black box neural network using the relationship between their VC
dimensions.

###  3.4 Tools and Techniques you intend to employ and develop

For the experimental part of our research, we will apply computer vision
architectures such as CNN’s to CIFAR-100 data, as well as create our own
synthetic data as in [11]’s paper. The CNN will serve as our complex deep
learning model which we wish to explain and study.

We will use pre-trained CNN’s like VGG-16 and feature extractors like
ScatterNet to both extract features for our boosted classifiers and measure
the “difficulty” of examples. Measuring the difficulty of an example will
involve finding a combination of how much it deviates from others in its class
and how close it is to other classes.

We intend to adopt proof techniques similar to [11], [8], and others in the
field. There, they precisely characterize the data distribution, architecture,
and training procedure. Common defaults for the setup involve a data
distribution with separable noise and signal components. A MLP with a single
hidden layer with hidden layer ReLU activation and sigmoid or tanh final
activation is commonly used for the architecture. Training procedure often
involves feature standardization and uses the SGD optimizer. For proofs, hinge
loss is often used.

We plan to use Adaboost due to its state-of-the-art results amongst boosting
algorithms and ease of implementation. We will use the same measure of mutual
information and plot this over the training time of our CNN expert model.

####  3.4.1 Experiment 1

We will train and checkpoint a typical CNN to greater than 90 percent
performance on CIFAR-10 first 5 versus last 5 binary classification, as well
as on synthetic high dimensional sinusoidal data where we can control the
ratio of signal to noise. Next, we will train and checkpoint adaboost at each
round of boosting, where we experiment with standard weak learners and
extracted features from VGG-16, as well as shallow CNN’s on the raw images as
in [16]. We can qualitatively examine the plots of mutual information as in
figure 1 to see observed phase separation.

Figure 1: The plots generated by [11] to qualitatively examine phase
separation

Our quantitative analysis will involve checking that both quantities

$I(G_{i};Y|F_{i+1})\text{ and }.I(F_{i};Y|G_{i+1})$ (3)

are close to zero. We will optimize the number of training epochs at which
each $F_{i}$ is defined for these two quantities to be small. Next, will
measure the statistical significance of the phase separation by averaging over
the quantities for many random classifiers of the same accuracy as $G_{i}$. To
construct a random classifier with accuracy $a$ we output the correct label
with probability $a$ and the incorrect label otherwise.

####  3.4.2 Experiment 2

Let errors($h_{j}$) and errors($f_{j}$) denote the emprical distribution of
errors (e.g. $|f_{j}(x)-y|\quad\forall(x,y)\in X\times Y$) over the dataset
using the sub-classifier $f_{j}$ directly to predict $X$. Aside from showing
the correlation between f and h to decrease per our theoretical analysis,
we’re also interested in looking at examples they make the same errors on.
Specifically, we expect the distribution of errors to be similar between the
NN sub-classifier $f_{j}$ and the weak classifier $h_{j}$ it matches with, and
the distribution of errors to diverge between two different sub-classifiers
$f_{j1}$ and $f_{j2}$, whose votes become more independent. Specifically, we
can show that after phase $i$, there is a sudden decrease in
$D_{KL}$(errors($f_{j}$), errors($h_{j}$)) and a sudden increase in
$D_{KL}$(errors($f_{j1}$), errors($f_{j2}$)).

Lastly, we make sure the final test error of a NN constrained this way still
compares favorably to a normally trained NN.

###  3.5 Initial Ideas

####  3.5.1 Theorem 1

We think we can inductively establish the result, given two assumptions.
First, $H^{T}$ is expressive, i.e. the data distribution and $H$ are suitably
chosen so $\operatorname*{arg\,max}_{f\in
H^{T}}l(f(X),Y)\approx\operatorname*{arg\,max}_{f\in F}l(f(X),Y)$ where $l$
denotes the likelihood objective. Second, the batch used to train $F_{i}$ in
phase $i$ is drawn from the distribution $S_{i}$, which was used to fit weak
classifier $i$. This will consist of the base case, where we show the neural
network learns a low-complexity weak learner, drawn from some hypothesis class
$H$. [11] among others already show this for $H$ being the full linear
classifier. Our base case will be no more difficult than that, since we sample
the batch from the same observation weight distribution chosen by Adaboost. A
proof sketch will be to show that $I(F_{i};Y|G_{i+1})\simeq 0\cap
F_{i+1}\leftarrow\text{train}(F_{i},S_{i})\Rightarrow
I(F_{i+1};Y|G_{i+2})\simeq 0$ and similarly $I(G_{i};Y|F_{i+1})\simeq 0\cap
F_{i+1}\leftarrow\text{train}(F_{i},S_{i})\Rightarrow
I(G_{i+1};Y|F_{i+2})\simeq 0$. Intuitively, $F_{i}$ and $h_{i}$ will minimize
risk on the same examples. In Theorem 2, we will further extend this
correspondence by finding a matching between $F_{i}$’s ”sub-classifiers” and
the weak classifiers $h_{j}$’s.

####  3.5.2 Theorem 2

In the neuron view of the sub-classifier, we think we can create a
correspondence between each neuron $f_{j}$ to a weak classifier $h_{j}$ by
adopting the arguments in [4], which shows a single step of gradient descent
can create ”neuron alignment”: a substantial number of neurons will correlate
to some cluster mean in their chosen xor-like data distribution, hence
activating on examples chosen from that cluster. They also show ”almost-
orthogonality”: an aligned neuron won’t activate on other cluster means (i.e.
the complement). An idealized interpretation of their result in our case is
that $x\in S_{j}\Leftrightarrow f_{j}(x)\approx 1$ and $x\not\in
S_{j}\Leftrightarrow f_{j}(x)\approx 0$.

In the subnetwork view of the sub-classifier, we think we can adopt a similar
approach, but analysis may call for some additional assumptions. Instead of
choosing an ensemble of subnetworks at random (similar to what dropout will
do), we impose we get to pick them such that the number of shared neurons
between any pair of them is small. By limiting to a small collection of
subnetworks, and training each on different sets of observation weights
informed by Adaboost, we can create a 1:1 correspondence between a subnetwork
and a weak classifier. Extending the assumption from Theorem 1, each sub-
classifier $f_{j}$ or $f^{J_{j}}$ will be trained batches sampled from
$S_{j}$, so it learns using the same observations as a corresponding weak
classifier, specializing to the same region.

Then, we will combine the aligned neurons (or ensemble of sub-networks) into a
single sub-network. [4] shows this creates a large margin sub-network (the
substantial number of aligned neurons comprising its hidden layer) which
results in a large margin classifier overall. This large margin sub-network’s
decision implicitly averages over these sub-classifiers’ votes, hence it has
the same self-averaging explanation as a boosted classifier!

####  3.5.3 Conjecture 1

To really make the connection hold in a wide variety of settings, we want to
find $G$ given $F$ for which both hypotheses hold. Our theoretical analysis
has assumed so far Adaboost yields a near-optimal classifier $G$ for which we
then constrain the training of $F$ in order for it to be explained by $G$ both
from correlation and causation. However, to truly deploy this into the real
world, we would need to construct both in parallel, and this conjecture
formulates our intuition that there’s a mapping between their complexities so
that given one, we can narrow down the search for the other.

####  3.5.4 Summary

Whereas there is little work explaining both the complexity and performance
gains of neural networks with every layer added, there is plenty of literature
on VC dimension bounds and generalization error guarantees for boosting from
PAC learning like [5], [2] and much more. We want to see if the claims of [11]
still hold when a linear classifier is swapped with boosted classifiers and
with VC dimension as the notion of complexity.

We propose to first establish an explanation using correlation (using a mutual
information based correlation measure through phases of training), then an
explanation using causation by a shared self-averaging mechanism over sub-
classifiers. We give experimental setups that can put these falsifiable
hypotheses to the test. If true, that can go a long way in uniting two
traditionally separate fields of research – boosting and deep learning – and
helping to ground the success of neural networks in more explored territory.
As a future direction, we envision when the situation calls for it, we can
readily swap a neural network black box decision with an alternative that
takes a small step down in accuracy but a giant leap forward in being
explainable in the features and interpretable in their decisions.

For additional visualizations and toy cases that complement this proposal, one
is welcome to reference our recent presentation.

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# Variable white dwarfs in TMTS: Asteroseismological analysis of a ZZ Ceti
star, TMTS J17184064+2524314

Jincheng Guo1,∗, Yanhui Chen2,3,4, Yonghui Yang2,3,4,∗, Xiaofeng Wang5,1,∗,
Jie Lin6,7,5, Xiao-Yu Ma8,9,Gaobo Xi5, Jun Mo5, Alexei V. Filippenko10, Thomas
G. Brink10, Weikai Zong8,9, Huahui Yan11,12,Jingkun Zhao12,13, Xiangyun
Zeng14, Zhihao Chen5, Ali Esamdin15, Fangzhou Guo5,Abdusamatjan Iskandar15,
Xiaojun Jiang12,13, Wenxiong Li16, Cheng Liu1, Jianrong Shi12,13,Xuan Song1,
Letian Wang15, Danfeng Xiang5, Shengyu Yan5, and Jicheng Zhang9
1Department of Scientific Research, Beijing Planetarium, Beijing 100044, China
2Institute of Astrophysics, Chuxiong Normal University, Chuxiong 675000, China
3International Centre of Supernovae (ICESUN), Yunnan Key Laboratory, Kunming
650216, China
4Faculty of Science, Kunming University of Science and Technology, Kunming
650093, China
5Physics Department and Tsinghua Center for Astrophysics, Tsinghua University,
Beijing 100084, China
6CAS Key laboratory for Research in Galaxies and Cosmology, Department of
Astronomy,
University of Science and Technology of China, Hefei, 230026, People’s
Republic of China
7School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei, 230026, People’s Republic of China
8Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, P. R. China
9Department of Astronomy, Beijing Normal University, Beijing 100875, China
10Department of Astronomy, University of California, Berkeley, CA 94720-3411,
USA
11Shandong Provincial Key Laboratory of Optical Astronomy and Solar-
Terrestrial Environment,
School of Space Science and Physics, Shandong University, Weihai 264209, China
12Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, China
13School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China
14Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang, 443000, People’s Republic of China
15Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi
830011, China
16The School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978,
Israel
∗ E-mail<EMAIL_ADDRESS>(J. Guo<EMAIL_ADDRESS>(Y.Yang<EMAIL_ADDRESS>(X. Wang)

###### Abstract

The Tsinghua University-Ma Huateng Telescope for Survey (TMTS) has been
constantly monitoring the northern sky since 2020 in search of rapidly
variable stars. To find variable white dwarfs (WDs), the TMTS catalog is
cross-matched with the WD catalog of Gaia EDR3, resulting in over 3000 light
curves of WD candidates. The WD TMTS J17184064+2524314 (hereafter J1718) is
the second ZZ Ceti star discovered among these common sources. Based on the
light curves from TMTS, follow-up photometric observations, and TESS, 10
periods and 3 combination periods are detected. A rotation period of $25.12\pm
0.18$ hr is derived, according to the identified rotational splitting. Our
spectroscopic observation indicates that this WD belongs to DA type with
$T_{\rm eff}=11,670\pm 604$ K, log $g=8.16\pm 0.36$, $M=0.70\pm 0.23$ M⊙, and
age=$0.51\pm 0.34$ Gyr. Based on core-parameterized asteroseismological model
grids ($\geqslant$ 14 million), we derive a best-fit solution of $T_{\rm
eff}=11,640\pm 20$ K, log $g=8.267\pm 0.008$, and $M=0.750\pm 0.005$ M⊙ for
J1718, consistent with the spectral fitting results. For this WD, the
corresponding carbon and oxygen abundances in the core are 0.43 and 0.57,
respectively. The distance derived from the intrinsic luminosity given by
asteroseismology is $64\pm 15$ pc, in accord with the distance of $70.1\pm
0.2$ pc from Gaia DR3 within the uncertainties.

###### keywords:

stars: white dwarfs – stars: variables: general – stars: oscillations – stars:
individual: TMTS J17184064+2524314

††pagerange: Variable white dwarfs in TMTS: Asteroseismological analysis of a
ZZ Ceti star, TMTS J17184064+2524314–LABEL:lastpage††pubyear: 2023

##  1 Introduction

Main-sequence stars with masses less than 10 M⊙ (Woosley & Heger, 2015),
depending on metallicity, will end up as white dwarfs (WDs). WDs can provide
information of great importance in various research areas (Fontaine &
Brassard, 2008; Winget & Kepler, 2008; Althaus et al., 2010; Saumon, Blouin, &
Tremblay, 2022). Because of their simple cooling mechanism, it is not
difficult to derive relatively accurate ages of WDs. This makes them suitable
for studies on the ages of Galactic populations (Kilic et al., 2019; Guo et
al., 2019), which are important tools for studying their mass functions
(Holberg et al., 2016; Hollands et al., 2018) and luminosity functions (Munn
et al., 2017; Lam et al., 2019). Accurate WD luminosity functions can be
further used to probe the structure and evolution of the Galactic disk and
open/globular clusters (Bedin et al., 2009; Campos et al., 2016; García-Berro
& Oswalt, 2016; Kilic et al., 2017; Guo et al., 2018). Additionally, as
potential gravitational-wave sources, ultrashort-period double WDs are likely
to be detected by space gravitational-wave facilities like LISA (Burdge et
al., 2019). Massive carbon-oxygen (CO) WDs in a binary system are the
progenitors of Type Ia supernovae (SNe Ia), which are standardizable candles
for research in cosmology (e.g., Maoz, Mannucci, & Nelemans, 2014).

Studies of WDs have entered a new era promoted greatly by the increased
numbers of stars (Kleinman et al., 2013; Kepler et al., 2015, 2019; Zhao et
al., 2013; Gentile Fusillo et al., 2015; Guo et al., 2015b, 2022; Kong & Luo,
2021) from large spectroscopic surveys like Palomar-Green (Green, Schmidt, &
Liebert, 1986), SDSS (York et al., 2000), and LAMOST (Zhao et al., 2012). But
thanks to the Gaia mission (Gaia Collaboration, et al., 2016), studies of WDs
now step into another new era. The number of WD candidates reached $\sim
359,000$ after the release of early Gaia DR3 (Gentile Fusillo et al., 2021).
These candidates are extremely useful for studies of photometric variability
of WDs.

Asteroseismology can be used to study the internal structure of pulsating
stars, including pulsating WDs (see a recent review by Córsico et al. (2019)).
Ever since the discovery of HL Tau 76 (Landolt, 1968), the first pulsating
hydrogen-rich WDs (DAV, or ZZ Ceti star), almost 500 DAVs have been identified
via photometric observations (Córsico, 2022; Guo et al., 2023), including
those discovered with TESS (Romero et al., 2022). Many of these DAVs have been
analysed with asteroseismology from different groups using different
approaches and models. Two methods are generally used: (i) a set of fully
evolutionary models, which are produced by tracking the complete evolution of
the progenitor stars, from the zero-age main sequence to the WD stage (Romero
et al., 2012, 2013; Córsico et al., 2013; De Gerónimo et al., 2017, 2018); and
(ii) a set of static stellar models with parameterized chemical profiles,
allowing a full exploration of the parameter space to locate an optimal
seismic model, leading to good matches to the observed periods (Bradley, 1998,
2001; Castanheira & Kepler, 2009; Bischoff-Kim, Montgomery, & Winget, 2008;
Bischoff-Kim & Metcalfe, 2011; Bischoff-Kim et al., 2014; Giammichele et al.,
2017a, b; Guo et al., 2023). Generally, the DAV instability strip ranges from
$\sim 10,800$ K to $\sim 12,000$ K (Gianninas, Bergeron, & Ruiz, 2011; Van
Grootel et al., 2012). The atmospheres of these cool WDs are dominated by
almost pure hydrogen, as a result of gravitational settling of other heavier
elements. The hydrogen in the outer envelope recombines at $T_{\rm eff}\approx
12,000$ K, resulting in a huge increase in envelope opacity. Accordingly, this
restrains the flow of radiation and eventually causes $g$-mode oscillations.
The excitation of $g$-mode oscillations in DA WDs at the blue edge of the
instability strip could be due to the $\kappa$ mechanism, i.e. partial
ionization of H (Winget et al., 1982). But a more physically plausible
explanation is the convective driving mechanism (Brickhill, 1991; Goldreich &
Wu, 1999) when the outer convection zone deepens due to cooling. The
oscillation-induced pulsation modes allow diagnosis of internal structures of
these stars, based on the fact that they are quite sensitive to the stellar
structure of DAVs.

High-precision asteroseismology of WDs (Østensen et al., 2011; Hermes et al.,
2017) began more than a decade ago (Hermes et al., 2011; Greiss et al., 2016)
with the Kepler mission (Borucki et al., 2010) and the K2 mission (Howell et
al., 2014). The sample has since been enriched by the TESS mission (Ricker et
al., 2015), which monitors the entire sky (Campante et al., 2016; Romero et
al., 2022). However, ground-based survey telescopes with short cadence are
still essential for discovery and asteroseismological study of faint pulsating
WDs.

In this work, we report the results of our asteroseismological analysis of the
second DAV star (TMTS J17184064+2524314, dubbed as J1718) from TMTS after TMTS
J23450729+5813146 (Guo et al., 2023). This star, with J2000 coordinates of
$\alpha=17^{\rm hr}18^{\rm m}40.64^{\rm s}$ and
$\delta=+25^{\circ}24^{\prime}31.4^{\prime\prime}$, was first reported as a WD
candidate using SDSS DR7 photometry (Girven et al., 2011). The spectral energy
distribution (SED) fitting suggests its effective temperature is 11,000 K and
surface gravity is 7.75. Similar results ($T_{\rm eff}=11,250$ K, log
$g$=7.50) were reported by Jiménez-Esteban et al. (2018) by fitting the SED as
well. Using accurate parameters from Gaia DR2, J1718 was found to be a thin-
disk star by Torres et al. (2019). The Gaia DR2 WD catalogue (Gentile Fusillo
et al., 2019) includes the estimated parameters of J1718 as $T_{\rm
eff}=11,595$ K, log $g=8.05$, and $M=0.64$ M⊙ for hydrogen-rich atmosphere
models, and $T_{\rm eff}=11,633$ K, log $g=7.98$, and $M=0.57$ M⊙ for helium-
rich atmosphere models. It was then studied in the Gaia 100 pc WD sample
(Kilic et al., 2020), where the estimation of parameters based on follow-up
spectroscopy suggests that J1718 is a DA WD with $T_{\rm eff}=11,361\pm 144$
K, log $g=8.04\pm 0.01$, and $M=0.63\pm 0.01$ M⊙.

J1718 was first identified as one of many new ZZ Ceti WDs by Vincent,
Bergeron, & Lafrenière (2020). According to two photometric follow-up
observations lasting 5 hr total, a main pulsation period of 731 s was
identified. Recently, variable WDs from Gaia and ZTF were studied by Guidry et
al. (2021). J1718 was observed by the McDonald 2.1 m telescope, where three
periods of 494.0 s, 397.3 s, and 531.9 s were detected. Moreover, three (495.2
s, 397.5 s, 532.1 s) and two (495.2 s, 401.2 s) periods were detected using
data from the ZTF $g$ and $r$ bands, respectively.

This paper is organized as follows. Sect. 2 presents the observational details
of J1718. In Sect. 3, frequency solution and asteroseismological modeling of
J1718 are presented. Discussions and a summary are given in Sect. 4.

##  2 Observations

###  2.1 TMTS

Located at Xinglong station in China, TMTS is a photometric survey with four
0.40 m optical telescopes. With a total field of view of $\sim 18$ deg2 (4.5
deg2 $\times 4$), it is equipped with four 4k $\times$ 4k pixel CMOS cameras,
achieving a read-out time of $<1$ s and high-speed photometry. Over the past
three years, the main survey mode conducted on TMTS is the uninterrupted
observations of LAMOST (Cui et al., 2012) sky areas for the whole night with a
cadence of 1 min, which allows the TMTS to coordinate with the spectroscopic
survey from the LAMOST observations. In order to fully exploit its potential,
a customized filter covering 3300–9000 Å is used for the survey. The 3$\sigma$
detection limit of TMTS can reach 19.4 mag for a 1 min exposure. More
information regarding the performance and research of TMTS can be found in
Zhang et al. (2020), Lin et al. (2022), Lin et al. (2023a, b), Guo et al.
(2023), and Liu et al. (2023).

Since 2020, the TMTS has produced uninterrupted whole-night light curves for
nearly 20 million objects. The WD catalog generated from Gaia EDR3 (Gentile
Fusillo et al., 2021) was cross-matched with the TMTS catalog within
$3^{\prime\prime}$, yielding TMTS light curves for almost 3000 WD candidates.
Based on visual inspections, J1718 was selected as a pulsating WD candidate
for further study. It was observed by TMTS on two nights (see details in Table
1), with the amplitude of variation being 0.2–0.3 mag, which is large for a ZZ
Ceti (similar to J2345).

Table 1: Observation logs of J1718. Telescope | Instrument | UTC date | Exp. (s)
---|---|---|---
TMTS | CMOS | 2020-05-09 | 60
|  | 2020-06-13 | 60
SNOVA | CCD | 2022-07-20 | 50
(0.4 m) |  | 2022-08-17 | 60
|  | 2022-08-18 | 60
TESS | 600-1000 nm | 2022-05-18 to | 20/120
|  | 2022-06-13/Sector 52 |
Shane | Kast | 2022-08-05 | 1800
(3 m) | spectrograph |  |
Figure 1: Part of the unfiltered light curve of SNOVA obtained in 2022.

Figure 2: Upper panel: A spectrum taken by the Lick 3 m Shane telescope (Kast
spectrograph) on Aug. 05, 2022. Lower panel: The spectral fits to the
normalized H${\alpha}$, H${\beta}$, H${\gamma}$, and H${\delta}$ lines. The
black lines represent the observed spectra, while the red dashed lines
represent the best model fitting.

###  2.2 Spectral observations

A spectrum was taken on August 05, 2022 (UTC dates are used throughout this
paper) with the Kast double spectrograph (Miles, & Stones, 1993) mounted on
the 3 m Shane telescope at Lick Observatory. The data reduction with
IRAF111IRAF: the Image Reduction and Analysis Facility is distributed by the
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy, Inc. (AURA) under cooperative
agreement with the National Science Foundation (NSF). follows standard
procedure, including dark-current subtraction, bias subtraction, cosmic-ray
removal, and one-dimensional (1D) spectral extraction. Following the spectral
model fitting approach described by Guo et al. (2015b, 2022), the parameters
of J1718 are derived to be $T_{\rm eff}=11,670\pm 604$ K, log $g=8.16\pm
0.36$, $M=0.70\pm 0.23$ M⊙, and cooling age $0.51\pm 0.34$ Gyr. The observed
Balmer line profiles and the best-fit models are shown in Figure 2. The
derived $T_{\rm eff}$ and mass are especially helpful for providing an
independent check with the results from asteroseismological analysis.

###  2.3 Photometric follow-up observations

Follow-up photometry of J1718 was obtained by using SNOVA, a 0.4 m telescope
located at Nanshan Observatory in China. The continuous observations were
obtained on three nights in white light (i.e., unfiltered) with an exposure
time of $\sim 50$–60 s. Standard image processing, such as bias correction,
flat correction, and source extraction are performed with IRAF. Differential
photometry was then applied to acquire the final light curves. Figure. 1
displays part of the light curve taken by SNOVA. The light variation of J1718
is $\sim 0.2$–0.3 mag, relatively large for a DAV. In addition to SNOVA, TESS
observed J1718 in sector 52 from May 18 to June 13, 2022 (see Table 1).

##  3 Analysis

Figure 3: The Lomb-Scargle periodogram of J1718 for 50 s and 60 s observations
by SNOVA and 20 s exposures by TESS in 2022. The horizontal blue dashed lines
are the 4.0$\sigma$ threshold. The red lines are identified frequencies above
4.0$\sigma$, while the green dashed lines are identified frequencies below but
close to the 4.0$\sigma$ threshold. The upside-down grey curves represent the
residuals of power spectra with significant frequencies being prewhitened.
Figure 4: The zoom-in Lomb-Scargle periodogram of J1718 for 20 s exposures by
TESS in 2022. The horizontal red dashed line is the 4.0$\sigma$ threshold. The
red text arrows are identified frequencies above 4.0$\sigma$, while the green
text arrows are identified frequencies below but close to the 4.0$\sigma$
threshold. Details of three possible splitting frequency regions are shown in
separate inset figures. Table 2: The frequency content detected from TESS 20 s
photometric observations.

ID | Freq. | $\sigma{\rm Freq.}$ | $P$ | $\sigma P$ | Amplitude | $\sigma\rm{Amp}$ | SNR
---|---|---|---|---|---|---|---
| ($\mu$Hz) | ($\mu$Hz) | (s) | (s) | (ppt) | (ppt) |
$f_{01}$ | 1238.570 | 0.027 | 807.383 | 0.017 | 17.102 | 1.755 | 9.74
$f_{02}$ | 1293.318 | 0.057 | 773.205 | 0.034 | 8.068 | 1.766 | 4.57
$f_{03}$ | 1304.248 | 0.013 | 766.725 | 0.008 | 34.231 | 1.751 | 19.55
$f_{04}$ | 1635.437 | 0.045 | 611.457 | 0.017 | 9.593 | 1.635 | 5.87
$f_{05}$ | 1646.616 | 0.024 | 607.306 | 0.009 | 17.569 | 1.633 | 10.76
$f_{06}^{*}$ | 1878.783 | 0.066 | 532.259 | 0.019 | 5.814 | 1.465 | 3.97
$f_{07}$ | 2010.509 | 0.059 | 497.386 | 0.015 | 6.564 | 1.486 | 4.42
$f_{08}$ | 2019.118 | 0.037 | 495.266 | 0.009 | 10.531 | 1.475 | 7.14
$f_{09}^{*}$ | 2835.372 | 0.070 | 352.687 | 0.009 | 5.645 | 1.522 | 3.71
$f_{10}^{*}$ | 3469.384 | 0.066 | 288.236 | 0.005 | 5.980 | 1.511 | 3.96
$f_{01}+f_{03}$ | 2542.866 | 0.061 | 393.257 | 0.009 | 6.754 | 1.580 | 4.28
$f_{03}+f_{03}$ | 2608.428 | 0.043 | 383.373 | 0.006 | 9.147 | 1.517 | 6.03
$f_{03}+f_{05}$ | 2950.704 | 0.064 | 338.902 | 0.007 | 6.366 | 1.571 | 4.05

  * •

∗ Frequencies below the 4.0$\sigma$ detection limits.

Table 3: Period distribution in J1718. (Columns 2 and 4 are frequencies and their corresponding amplitudes, respectively. Column 3 is the frequency difference between split frequencies.) ID | Freq |  $\delta$ $f$ | A | $P$ | $m$ | $l$
---|---|---|---|---|---|---
| ($\mu$Hz) | ($\mu$Hz) | (ppt) | (s) |  |
$f_{01}$ | 1238.570 |  | 17.102 | 807.383 | 0? | 1 or 2
$f_{02}$ | 1293.318 |  | 8.068 | 773.205 | -1 | 1
|  | 5.465 |  |  |  |
| 1298.783 |  |  | 769.951 | 0 | 1
|  | 5.465 |  |  |  |
$f_{03}$ | 1304.248 |  | 34.231 | 766.725 | +1 | 1
$f_{04}$ | 1635.437 |  | 9.593 | 611.457 | -1 | 1
|  | 5.590 |  |  |  |
| 1641.027 |  |  | 609.374 | 0 | 1
|  | 5.590 |  |  |  |
$f_{05}$ | 1646.616 |  | 17.569 | 607.306 | +1 | 1
$f_{06}$ | 1878.783 |  | 5.814 | 532.259 | 0? | 1 or 2
$f_{07}$ | 2010.509 |  | 6.564 | 497.386 | -1 | 2
|  | 8.609 |  |  |  |
$f_{08}$ | 2019.118 |  | 10.531 | 495.266 | 0 | 2
$f_{09}$ | 2835.372 |  | 5.645 | 352.687 | 0? | 1 or 2
$f_{10}$ | 3469.384 |  | 5.98 | 288.236 | 0? | 1 or 2
Table 4: Identification of the modes observed in J1718. The second column is the periods of our optimal model. $P_{\rm obs}$ | $P_{\rm model}$ | $l$ | $k$
---|---|---|---
(s) | (s) |  |
288.236 | 288.204 | 1 | 4
352.687 | 353.635 | 2 | 12
495.266 | 495.521 | 2 | 18
532.259 | 532.222 | 1 | 10
609.374 | 609.031 | 1 | 12
769.951 | 769.841 | 1 | 16
807.383 | 807.182 | 2 | 31

###  3.1 The amplitude spectrum and pulsation frequencies

For the purpose of Fourier decomposition of the light curve, the Frequency
Extraction for Light-curve exploitation (FELIX) software package was used
(Charpinet et al., 2010), which is dedicated to Kepler and TESS photometry
(Zong et al., 2016; Charpinet et al., 2019). This package includes standard
prewhitening and nonlinear least-squares fitting techniques for Fourier
transformation (Deeming, 1975; Scargle, 1982). The acceptance threshold for a
significant frequency is set to be 4.0$\sigma$ of the local noise level (e.g.,
Kuschnig et al., 1997).

Figure 4 shows the Lomb-Scargle periodograms of SNOVA (50 s and 60 s
exposures) and TESS (20 s exposure) observations; 5 and 13 frequencies were
detected in the SNOVA and TESS observations, respectively, including 2
frequencies in SNOVA and 3 frequencies in TESS below the detection limit.
Owing to the fact that five frequencies detected in SNOVA observations are all
included in the TESS frequencies, all 13 frequencies detected in TESS data are
used for further analysis. Table 2 lists the frequencies with their attributes
extracted with FELIX: the frequency ID (Column 1) ordered by their frequency
value (Column 2), amplitude (Column 6) and uncertainty (Columns 3, 7), the
corresponding period and uncertainty (Columns 4, 5), and the signal-to-noise
ratio (SNR, Column 8).

Among the 13 frequencies, 10 are significant and 3 are suspected (below
4.0$\sigma$, marked by the "$\ast$" symbol). According to our careful
examination, three linear combination frequencies are identified within their
uncertainties, which are listed in the last three rows in Table 2.

###  3.2 Period distribution

Based on Table 2, there are 10 independent signals left, after removing the
linear combination frequencies. Since frequencies $f_{02}$ and $f_{03}$,
$f_{04}$ and $f_{05}$, and $f_{07}$ and $f_{08}$ are very close, we can assume
that they are components of two different triplets and one quintuplet,
respectively. Reasonably, we can also assume that in the two triplets, two
central modes are missing between $f_{02}$ and $f_{03}$, as well as between
$f_{04}$ and $f_{05}$. Three modes are missing in the quintuplet, which
contains $f_{07}$ and $f_{08}$. In order to carry out further analysis, the
missing central modes of two triplets are averaged to be 1298.783 $\mu$Hz and
1641.027 $\mu$Hz. In this setting, they satisfy the relation that
$[(5.465\times 2)+(5.590\times 2)]/4/8.609=0.642\approx 0.6$. Meanwhile, there
are equally spaced frequencies in the first two triplets. Therefore, we
identify them as $l=1$ modes with the central frequencies of $m=0$ modes
undetected. The third incomplete triplet is assumed to be the $l=2$ mode. The
detailed mode identifications are listed in Table 3.

###  3.3 Rotational splitting

As shown in Table 3, there are two incomplete triplets identified as $l=1$
modes. They are interpreted as results of rotational splitting. An average
value of frequency separation between the nearby two components of the
triplets is derived as (5.465 + 5.465 + 5.590 + 5.590)/4 = 5.528 $\mu$Hz.
Considering the first order in the rotation angular frequency $\Omega$, the
frequencies in the rotating case $\sigma_{k,l,m}$ ($m\neq 0$) are associated
with the frequencies in the nonrotating case $\sigma_{k,l}$ ($m=0$) in the
relation

$\Omega=\frac{\left(\sigma_{k,l,m}-\sigma_{k,l}\right)}{m\times\left(1-C_{k,l}\right)}\,,$
(1)

where $C_{k,l}=1/l(l+1)$ is in the asymptotic regime (Brickhill, 1975). By
adopting this equation, the rotation period of J1718 is derived to be
$25.12\pm 0.18$ hr.

###  3.4 Input physics and model calculations

In order to evolve grids for DAV models, version 16 of the White Dwarf
Evolution Code (Bischoff-Kim, 2018) is applied. This code includes a new
feature that enables modeling of the oxygen (O) profile in the core, which is
parameterized by six parameters (i.e., $h1-h3$, $w1-w3$; see Fig. 7). This
feature provides more accurate fitting results than previous versions of the
code that did not include it. The equation of state and opacity tables from
the Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al.,
2018, 2019) are adopted in the current version of WDEC. The standard mixing-
length theory (Bohm & Cassinelli, 1971) and a mixing-length parameter of 0.6
(Bergeron et al., 1995) are adopted in the analysis.

The explored parameter spaces of evolved WD models are shown in Table 5. To
cover the initial ranges and crude steps in this table, a total of 14,486,688
WD models have been evolved. This is the first time that such a large number
of evolved model grids are used for DAV model fitting. This applied model grid
differs from those derived by considering the complete evolution of WDs in two
ways. First, this model grid only considers the cooling phase of WDs, while
fully evolutionary models take into account the complete evolution of the
progenitor stars and the WD cooling phases. Second, fully evolutionary models
are typically not used for model fitting of large samples, while the model
adopted here constructs a large grids to fit the observed periods. Besides,
this model includes parameterized central core as well. More description about
the core parameters can be found in Figure 1 of Bischoff-Kim (2018) and the
user manual (Bischoff-Kim & Montgomery, 2018). In the following fitting
process, fine steps are used to optimize the best-fitting models found with
crude steps.

Table 5: The explored parameter spaces of WDs evolved by WDEC.

Parameters | Initial ranges | Crude steps | Fine steps | Optimal values | 2nd Optimal values
---|---|---|---|---|---
$M_{*}$/M⊙ | [0.500,0.850] | 0.010 | 0.005 | 0.750$\pm$0.005 | 0.800$\pm$0.005
$T_{\rm eff}$ (K) | [10600,12600] | 250 | 10 | 11640$\pm$20 | 11090$\pm$10
$-$log($M_{\rm env}/M_{\rm*}$) | [1.50,2.00] | 0.50 | 0.01 | 2.00$\pm$0.01 | 1.50$\pm$0.01
| [2.00,3.00] | 1.00 | 0.01 |  |
$-$log($M_{\rm He}/M_{\rm*}$) | [2.00,5.00] | 1.00 | 0.01 | 3.00$\pm$0.01 | 2.00$\pm$0.01
$-$log($M_{\rm H}/M_{\rm*}$) | [4.00,10.00] | 1.00 | 0.01 | 6.00$\pm$0.01 | 8.00$\pm$0.01
$X_{\rm He}$ in mixed C/He/H region | [0.10,0.90] | 0.16 | 0.01 | 0.24$\pm$0.02 | 0.43$\pm$0.01
$X_{\mathrm{O}}$ in the core |  |  |  |  |
$h1$ | [0.60,0.75] | 0.03 | 0.01 | 0.57$\pm$0.02 | 0.59$\pm$0.01
$h2$ | [0.65,0.71] | 0.03 | 0.01 | 0.66$\pm$0.02 | 0.61$\pm$0.01
$h3$ | 0.85 |  | 0.01 | 0.83$\pm$0.03 | 0.90$\pm$0.01
$w1$ | [0.32,0.38] | 0.03 | 0.01 | 0.30$\pm$0.01 | 0.33$\pm$0.01
$w2$ | [0.42,0.48] | 0.03 | 0.01 | 0.49$\pm$0.02 | 0.42$\pm$0.01
$w3$ | 0.09 |  | 0.01 | 0.13$\pm$0.02 | 0.13$\pm$0.01

  * •

$M_{*}$ denotes the stellar mass, while $M_{\rm env}$ is the envelope mass.
$M_{\rm H}$ is the mass of the hydrogen atmosphere, and $M_{\rm He}$ is the
mass of the He layer. $X_{He}$ is the He abundance in the mixed C/He/H region,
and $X_{O}$ is the O abundance in the core. The parameter $h1$ refers to the O
abundance in the core center, while $w1$ is the mass fraction of $X_{O}=h1$.
Parameters $h2$ and $h3$ are O abundance of two knee points on the declining O
profile; $w2$ and $w3$ refer to the masses of the gradient regions of the O
profile.

Figure 5: Upper panel: a color 3D graph of the fitting error $\sigma_{\rm
RMS}$ as a function of stellar mass and $T_{\rm eff}$. A total of 14,271 DAV
stellar models are used in the fit. Lower panel: a color 3D graph of the
fitting error $\sigma_{\rm RMS}$ as a function of H-layer mass and envelope
mass. A total of 12,801 DAV stellar models are used in the fit.

###  3.5 The model fittings

To evaluate the fitting results, the root-mean-square (RMS) residual
$\sigma_{\rm RMS}$ is used,

$\sigma_{\rm RMS}=\sqrt{\frac{1}{n}\sum_{n}(P_{\rm obs}-P_{\rm mol})^{2}}\,,$
(2)

where $n$ represents the number of the fit modes (i.e., 7 for J1718). The
observed and model periods are denoted as $P_{\rm obs}$ and $P_{\rm mod}$,
respectively. During the model fittings, only the first 3 digits after the
decimal point of the observed periods are used. Because the periods 773.205 s
and 766.725 s, 611.457 s and 607.306 s, and 497.386 s and 495.266 s are very
close, we assume that they are components of two rotational triplets and one
quintuplet, respectively. Therefore, we use their central modes of 769.951 s,
609.374 s, and 495.266 s, respectively. Together with 4 other periods, they
are listed in the first column of Table 4 as the observed periods.

In the initial fitting, the combinations of parameter space ranges and crude
steps are used to build the DAV models. This includes 14,486,688 models, which
are adopted to fit the observed 7 modes. The crude steps are chosen by
considering the computational power for models. It takes about 7 s to evolve
one model using a 128-core/256-threads server with a clock speed of 2.45 GHz.
After finding the models with the minimum $\sigma_{\rm RMS}$, the parameter
steps of the models are narrowed down to fine steps around those models. For
the fine steps, there are 3 to 5 grid points for each parameter. The fine
steps of mass are chosen to be 0.005 because this is the step size adopted in
the WDEC. The fine steps of other parameters are chosen to be much smaller
than their crude steps. There are 6 global parameters ($M_{*}$/M⊙, $T_{\rm
eff}$, $-$log($M_{\rm env}/M_{\rm*}$), $-$log($M_{\rm He}/M_{\rm*}$),
$-$log($M_{\rm H}/M_{\rm*}$), and $X_{\rm He}$ in mixed C/He/H region) and 6
$X_{O}$ parameters ($h1-h3$, $w1-w3$) in Table 5.

In order to find the minimum $\sigma_{\rm RMS}$, the $X_{O}$ parameters are
fixed at first. Then by using fine steps, a grid of global parameters is
generated to fit. Next, the global parameters are fixed to previously found
values, and using fine steps, a grid of $X_{O}$ parameters is generated to
fit. This process is repeated a dozen times, and the chosen fitting model
converges to an optimal one.

In the actual fitting process, the optimal model using crude step grids is
found to be $T_{\rm eff}=11,850$ K and $M=0.85$ M⊙ with $\sigma_{\rm
RMS}=1.06$, while the second optimal model is obtained to be $T_{\rm
eff}=11,600$ K and $M=0.75$ M⊙ with $\sigma_{\rm RMS}=1.21$. Owing to the fact
that the mass of the second optimal model is more consistent with our
spectral-fitting mass (i.e., 0.70 M⊙), we chose to find the final optimal
model using fine steps based on the second optimal model. Then the final
optimal model using fine steps is obtained when $\sigma_{\rm RMS}=0.4$ s, with
the corresponding periods and other parameters listed in the second column of
Table 4 and fifth column of Table 5, respectively. The uncertainty of each
optimal value is estimated by the full width at half height of the reciprocal
of $n\sigma_{\rm RMS}$. According to the derived optimal values, J1718 is a
relatively massive DAV with a slightly thick H atmosphere.

Figure 6: Sensitivity plots of 8 parameters of the optimal model. Subplots are
8 parameter values versus the residuals of their RMS, $\sigma_{\rm RMS}$.
Figure 7: Chemical profile of oxygen as a function of the mass coordinate.
Blue dashed line is the oxygen abundance profile in the core described by six
parameters. Figure 8: The core composition profiles and corresponding Brunt-
Väisälä frequency of the optimal model as functions of the mass coordinate
logarithm. Upper panel: the logarithm of the square of buoyancy frequency as a
function of stellar mass. Several bumps are present due to the existence of
element transition gradient zones. Lower panel: the abundance of O, C, He, and
H, plotted in yellow, green, blue, and brown lines, respectively.

The upper panel of Fig. 5 shows a three dimensional (3D) color graph of the
fitting error $\sigma_{\rm RMS}$ as a function of stellar mass and $T_{\rm
eff}$. For the purpose of a better visual effect, the $z$-axis and colour
represent the inverse of the fitting error $\sigma_{\rm RMS}$ (as defined in
Eq.2) squared. The grid consists of stellar mass in the range of 0.500 M⊙ to
0.850 M⊙ with a step of 0.005 M⊙ and $T_{\rm eff}$ in the range of 10,600 K to
12,600 K with a step of 10 K, which contains 14,271 DAV models. All other
parameters are fixed to their optimal values in Table 5. The lower panel of
Fig. 5 shows $\sigma_{\rm RMS}$ as a function of hydrogen layer mass and
envelope mass, which involves 12,801 DAV models. The logarithm of the envelope
mass ranges from $-1.50$ to $-2.40$ with a step of 0.02, and the logarithm of
H layer mass ranges from $-5.00$ to $-10.00$ with a step of 0.02. The highest
points in both figures correspond to values of our optimal $T_{\rm eff}$,
mass, H-layer mass, and envelope mass.

Fig. 6 displays the sensitivity plots (fitting error $\sigma_{\rm RMS}$) of
eight parameters of the optimal model, including the stellar mass, $T_{\rm
eff}$, envelope mass, helium layer mass, hydrogen atmosphere mass, helium
abundance in the mixed C/He/H region, and $h1$ and $w1$ for the O profile.
Note that the sensitivity of each parameter is derived when all other
parameters are fixed to the optimal values. The results with the lowest
$\sigma_{\rm RMS}$ indicate the optimal values. In Table 5, the mass of the H
shell is in the range of $10^{-10}\,M_{\ast}$ to $10^{-4}\,M_{\ast}$. One
should note that after an iteration, the fitting process may explore a broader
range of values according to the pre-optimal solution. This is because we want
to find the optimal values for the parameters, even if they are outside of the
initial ranges listed in Table 5.

Fig. 7 is plotted according to six parameters, listed in Table 5, that
describe oxygen abundance in the core. The blue dashed line represents the
oxygen profile as a function of the mass coordinate. The definitions of the
six parameters are demonstrated in the figure. The core composition profiles
and corresponding Brunt-Väisälä frequency for the optimal model are shown in
Fig. 8. The bumps in the profile are caused by composition gradient in
different zones. It is important to note that this kind of chemical profile is
in contrast to those derived from the fully evolutionary computations. More
detailed information regarding the existence of a C-buffer and several
features in the chemical profile has been discussed extensively by De Gerónimo
et al. (2019) for a DBV star.

To check the self-consistency of the best-fit model, we can use the distance
derived from asteroseismology analysis to perform an independent check. The
luminosity of J1718 is log($L$/L$\odot$) $=-2.736\pm 0.183$. Using the
correction $M_{\rm bol}=M_{\rm bol,\odot}-2.5\times log(L/{\rm L}_{\odot})$
and assuming the bolometric magnitude of the Sun $M_{\rm bol,\odot}=4.74$
(Cox, 2000), the bolometric magnitude of the optimal model is $M_{\rm
bol}=11.58$ mag. The $V$-band bolometric correction (BC) is adopted for J1718,
as the Gaia $G$-band BC is not available in the DR3. As listed in Table 1 of
Bergeron, Wesemael, & Beauchamp (1995), the $V$-band BC for a DA star model
with $T_{\rm eff}=11,000$ K and 12,000 K is $-$0.441 mag and $-$0.611 mag,
respectively. Therefore, for J1718 with $T_{\rm eff}=11,640$ K, the linear
interpolation of BC ($V$) is $-$0.550 mag. Thus, the absolute $V$-band
magnitude is derived to be $M_{V}=M_{\rm bol}-BC(V)=11.58-(-0.550)=12.13$ mag.
Given $m_{V}=16.152$ mag (Qi et al., 2015), the distance inferred from the
optimal model is $64\pm 15$ pc by using the equation
$d=10^{(m_{V}-M_{V})/5+1}$. Within the quoted uncertainty, this result is
consistent with the parallax-inversed distance of $70.1\pm 0.2$ pc from the
Gaia DR3. The consistency of the asteroseismology analysis with that derived
from the spectral model fitting indicates that our asteroseismological results
are self-consistent.

###  3.6 Alternative solutions

In addition to the adopted solution, two more period interpretations are
explored as well. In one scenario, we consider that top 10 frequencies listed
in Table 2 are all independent frequencies without splitting. After repeating
the same period-fit process mentioned above are applied, the final optimal
model is obtained with fitting error $\sigma_{\rm RMS}$=1.26 s. However, the
parameters of $T_{\rm eff}$=12 530 K and mass=0.89 M⊙ differ a lot from
parameters derived from spectral fitting results of $T_{\rm eff}$=11 670 K and
mass=0.70 M⊙. Moreover, the derived asteroseismological distance is 59 pc,
which is 19% less than Gaia distance. And there is only one period, 496.687 s,
exists in the optimal model in the range of 400-500 s, while two periods of
495.266 s and 497.386 s are observed. Based on all the evidences, this
scenario is ruled out.

In the second scenario, we assume that there are 7 independent frequencies,
which are listed in Table 6. This time we consider that $f_{03}$, $f_{05}$ and
$f_{08}$ are central frequencies of three different triplets, and for each
triplet, one frequency is not detected in the TESS observation. After
repeating the period-fit analysis, the optimal model for this scenario is
found with $\sigma_{\rm RMS}$=0.09 s. The corresponding parameters are listed
in the last column of Table 5. The parameters of optimal model are consistent
with spectral fitting results within the uncertainties. But the derived
asteroseismological distance of this solution is 58 pc, which is 21% less than
Gaia distance of 70 pc. Furthermore, among the 7 observed independent
frequencies, there are four $l=2$ modes more than three $l=1$ modes in the
optimal solution. Usually, $l=1$ mode should be easier to be detected than
$l=2$ mode in an observation. So among the observed frequencies, it should be
normal to find more $l=1$ modes than $l=2$ modes. Therefore, we still prefer
the period interpretation in Table 3 and adopt the optimal values listed in
the fifth column of Table 5 as the final optimal solution.

Table 6: Alternative identification of the modes observed in J1718. The second column is the periods of 2nd optimal model. $P_{\rm obs}$ | $P_{\rm model}$ | $l$ | $k$
---|---|---|---
(s) | (s) |  |
288.236 | 288.161 | 2 | 9
352.687 | 352.775 | 2 | 12
495.266 | 495.144 | 1 | 9
532.259 | 532.162 | 2 | 19
607.306 | 607.334 | 1 | 12
766.725 | 766.692 | 1 | 16
807.383 | 807.268 | 2 | 30

##  4 Discussion and Summary

Although a few pulsating periods of the ZZ Ceti star J1718 were previously
detected, we performed a thorough asterosesimological analysis for the first
time with more extensive data, deriving accurate parameters for this pulsating
white dwarf.

Based on data from the TMTS and SNOVA, multiple pulsating periods were
identified after prewhitening. However, since J1718 was observed by TESS in
2022, whose pulsating periods covered all TMTS and SNOVA periods, we only
focus on the TESS-detected periods. There are 13 frequencies detected in TESS
20 s data after prewhitening. Among those, 3 frequencies are linear
combination frequencies. The remaining ten frequencies are divided into two
incomplete triplets, one incomplete quintuplet, and four independent
frequencies. The rotational period of $25.12\pm 0.18$ hr is derived using two
triplets listed in Table 3.

The identified modes are used to perform the asteroseismological analysis. By
adopting version 16 of WDEC, more than 14 million WD models were evolved for
the parameter spaces listed in Table 5. To find the optimal model, parameter
steps and spaces are slowly narrowed down repeatedly, in accordance to the
fitting result. Finally, the optimal model is found and its parameter values
are listed in Table 5. From the asteroseismological result, it can be learned
that J1718 is a relatively massive pulsating WD with a slightly thick H
atmosphere. The final results are consistent with the parameters derived from
spectral fitting of our follow-up spectrum. In addition, the distance deduced
from asteroseismological results is consistent with the inverse parallax
derived distance from Gaia DR3 within the uncertainty, implying relatively
good self-consistency. It is noted that there is a 9% distance discrepancy. We
tend to think that it could be caused by the large light-curve variation. The
apparent magnitude in the $V$ band of 16.152 is adopted here. However,
considering a 0.2–0.3 mag variation, $m_{V}$ could be 16.352 mag, which would
lead to a asteroseismological distance of $\sim 70$ pc, which equals the
astrometric distance of 70 pc from Gaia DR3.

## Acknowledgments

The authors thank the referee Dr. A. H. Córsico for valuable suggestions and
advices, which improved this work a lot. The authors acknowledge the National
Natural Science Foundation of China (NSFC) under grants 12203006, 12033003,
12288102, U1938113, 11773035, 12090044, 12090042, and 11633002. This work is
also supported by the Ma Huateng Foundation, the Scholar Program of Beijing
Academy of Science and Technology (DZ:BS202002), the Tencent Xplorer Prize,
and CSST projects: CMS-CSST-2021-B03, the Innovation Project of Beijing
Academy of Science and Technology (11000023T000002062763-23CB059 and
11000022T000000443055), and the International Centre of Supernovae, Yunnan Key
Laboratory (No. 202302AN36000101).

A.V.F.’s supernova group at UC Berkeley is grateful for financial support from
the Christopher R. Redlich Fund, Briggs and Kathleen Wood (T.G.B. is a Wood
Specialist in Astronomy), and numerous other donors. A major upgrade of the
Kast spectrograph on the Shane 3 m telescope at Lick Observatory, led by Brad
Holden, was made possible through generous gifts from the Heising-Simons
Foundation, William and Marina Kast, and the University of California
Observatories. Research at Lick Observatory is partially supported by a
generous gift from Google.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC;
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al.,
2000).

## DATA AVAILABILITY

The TESS data used in this article can be accessed via
https://archive.stsci.edu/missions-and-data/tess. The TMTS, SNOVA photometric,
and 3 m spectral data can be obtained by contacting the corresponding authors.

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