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The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Mat\'ern covariance functions are essential. However, the Mat\'ern covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Mat\'ern covariance function instead. First, we show that this circular Mat\'ern function is the covariance of the stationary solution to the stochastic differential equation on the circle with a formally defined white noise space measure. Then, for the corresponding conditional autoregressive model, we derive a closed form formula for its covariance function. Together with a closed form formula for the circular Mat\'ern covariance function, the link between these two random fields can be established explicitly. Additionally, it is known that the estimator of the mean is not consistent on circles, we provide an equivalent Gaussian measure explanation for this non-ergodicity issue.

Given a probability distribution $\mathcal{D}$ over the non-negative integers, a $\mathcal{D}$-repeat channel acts on an input symbol by repeating it a number of times distributed as $\mathcal{D}$. For example, the binary deletion channel ($\mathcal{D}=Bernoulli$) and the Poisson repeat channel ($\mathcal{D}=Poisson$) are special cases. We say a $\mathcal{D}$-repeat channel is square-integrable if $\mathcal{D}$ has finite first and second moments. In this paper, we construct explicit codes for all square-integrable $\mathcal{D}$-repeat channels with rate arbitrarily close to the capacity, that are encodable and decodable in linear and quasi-linear time, respectively. We also consider possible extensions to the repeat channel model, and illustrate how our construction can be extended to an even broader class of channels capturing insertions, deletions, and substitutions. Our work offers an alternative, simplified, and more general construction to the recent work of Rubinstein (arXiv:2111.00261), who attains similar results to ours in the cases of the deletion channel and the Poisson repeat channel. It also improves on the decoding failure probability of the polar codes constructions of Tal et al. for the deletion channel (ISIT 2019) and certain insertion/deletion/substitution channels (arXiv:2102.02155). Our techniques follow closely the approaches of Guruswami and Li (IEEEToIT 2019) and Con and Shpilka (IEEEToIT 2020); what sets apart our work is that we show that a capacity-achieving code for the channels in question can be assumed to have an "approximate balance" in the frequency of zeros and ones of all sufficiently long substrings of all codewords. This allows us to attain near-capacity-achieving codes in a general setting. We consider this "approximate balance" result to be of independent interest, as it can be cast in much greater generality than repeat channels.

The issues of bias-correction and robustness are crucial in the strategy of divide-and-conquer (DC), especially for asymmetric nonparametric models with massive data. It is known that quantile-based methods can achieve the robustness, but the quantile estimation for nonparametric regression has non-ignorable bias when the error distribution is asymmetric. This paper explores a global bias-corrected DC by quantile-matched composite for nonparametric regressions with general error distributions. The proposed strategies can achieve the bias-correction and robustness, simultaneously. Unlike common DC quantile estimations that use an identical quantile level to construct a local estimator by each local machine, in the new methodologies, the local estimators are obtained at various quantile levels for different data batches, and then the global estimator is elaborately constructed as a weighted sum of the local estimators. In the weighted sum, the weights and quantile levels are well-matched such that the bias of the global estimator is corrected significantly, especially for the case where the error distribution is asymmetric. Based on the asymptotic properties of the global estimator, the optimal weights are attained, and the corresponding algorithms are then suggested. The behaviors of the new methods are further illustrated by various numerical examples from simulation experiments and real data analyses. Compared with the competitors, the new methods have the favorable features of estimation accuracy, robustness, applicability and computational efficiency.

We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent $\mu$ of the family of codes. The scaling exponent has emerged as a central object in the finite-length characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is $O(n^{-1/\mu})$, and (ii) this upper bound can be improved to roughly $O(n^{-1/2})$ by considering polar codes with large kernels. We conjecture that these bounds could be improved to $O(n^{-2/\mu})$ and $O(n^{-1})$, respectively, and provide a heuristic argument as well as numerical evidence supporting this view.

A new source model, which consists of an intrinsic state part and an extrinsic observation part, is proposed and its information-theoretic characterization, namely its rate-distortion function, is defined and analyzed. Such a source model is motivated by the recent surge of interest in the semantic aspect of information: the intrinsic state corresponds to the semantic feature of the source, which in general is not observable but can only be inferred from the extrinsic observation. There are two distortion measures, one between the intrinsic state and its reproduction, and the other between the extrinsic observation and its reproduction. Under a given code rate, the tradeoff between these two distortion measures is characterized by the rate-distortion function, which is solved via the indirect rate-distortion theory and is termed as the semantic rate-distortion function of the source. As an application of the general model and its analysis, the case of Gaussian extrinsic observation is studied, assuming a linear relationship between the intrinsic state and the extrinsic observation, under a quadratic distortion structure. The semantic rate-distortion function is shown to be the solution of a convex programming programming with respect to an error covariance matrix, and a reverse water-filling type of solution is provided when the model further satisfies a diagonalizability condition.

In recent years, understanding the implicit regularization of neural networks (NNs) has become a central task of deep learning theory. However, implicit regularization is in itself not completely defined and well understood. In this work, we make an attempt to mathematically define and study the implicit regularization. Importantly, we explore the limitation of a common approach of characterizing the implicit regularization by data-independent functions. We propose two dynamical mechanisms, i.e., Two-point and One-point Overlapping mechanisms, based on which we provide two recipes for producing classes of one-hidden-neuron NNs that provably cannot be fully characterized by a type of or all data-independent functions. Our results signify the profound data-dependency of implicit regularization in general, inspiring us to study in detail the data-dependency of NN implicit regularization in the future.

A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.

The phenomenon of benign overfitting, where a predictor perfectly fits noisy training data while attaining low expected loss, has received much attention in recent years, but still remains not fully understood beyond simple linear regression setups. In this paper, we show that for regression, benign overfitting is "biased" towards certain types of problems, in the sense that its existence on one learning problem excludes its existence on other learning problems. On the negative side, we use this to argue that one should not expect benign overfitting to occur in general, for several natural extensions of the plain linear regression problems studied so far. We then turn to classification problems, and show that the situation there is much more favorable. Specifically, we consider a model where an arbitrary input distribution of some fixed dimension $k$ is concatenated with a high-dimensional distribution, and prove that the max-margin predictor (to which gradient-based methods are known to converge in direction) is asymptotically biased towards minimizing the expected *squared hinge loss* w.r.t. the $k$-dimensional distribution. This allows us to reduce the question of benign overfitting in classification to the simpler question of whether this loss is a good surrogate for the prediction error, and use it to show benign overfitting in some new settings.

Recent works have shown explainability and robustness are two crucial ingredients of trustworthy and reliable text classification. However, previous works usually address one of two aspects: i) how to extract accurate rationales for explainability while being beneficial to prediction; ii) how to make the predictive model robust to different types of adversarial attacks. Intuitively, a model that produces helpful explanations should be more robust against adversarial attacks, because we cannot trust the model that outputs explanations but changes its prediction under small perturbations. To this end, we propose a joint classification and rationale extraction model named AT-BMC. It includes two key mechanisms: mixed Adversarial Training (AT) is designed to use various perturbations in discrete and embedding space to improve the model's robustness, and Boundary Match Constraint (BMC) helps to locate rationales more precisely with the guidance of boundary information. Performances on benchmark datasets demonstrate that the proposed AT-BMC outperforms baselines on both classification and rationale extraction by a large margin. Robustness analysis shows that the proposed AT-BMC decreases the attack success rate effectively by up to 69%. The empirical results indicate that there are connections between robust models and better explanations.

Adapting semantic segmentation models to new domains is an important but challenging problem. Recently enlightening progress has been made, but the performance of existing methods are unsatisfactory on real datasets where the new target domain comprises of heterogeneous sub-domains (e.g., diverse weather characteristics). We point out that carefully reasoning about the multiple modalities in the target domain can improve the robustness of adaptation models. To this end, we propose a condition-guided adaptation framework that is empowered by a special attentive progressive adversarial training (APAT) mechanism and a novel self-training policy. The APAT strategy progressively performs condition-specific alignment and attentive global feature matching. The new self-training scheme exploits the adversarial ambivalences of easy and hard adaptation regions and the correlations among target sub-domains effectively. We evaluate our method (DCAA) on various adaptation scenarios where the target images vary in weather conditions. The comparisons against baselines and the state-of-the-art approaches demonstrate the superiority of DCAA over the competitors.