In the study of sparse stochastic block model (SBM) one needs to analyze a distributional recursion, known as belief propagation (BP) on a tree. Uniqueness of the fixed point of this recursion implies several results about the SBM, including optimal recovery algorithms for SBM (Mossel et al. (2016)) and SBM with side information (Mossel and Xu (2016)), and a formula for SBM mutual information (Abbe et al. (2021)). The 2-community case corresponds to an Ising model, for which Yu and Polyanskiy (2022) established uniqueness for all cases. Here, we analyze broadcasting of $q$-ary spins on a Galton-Watson tree with expected offspring degree $d$ and Potts channels with second-largest eigenvalue $\lambda$. We allow for the intermediate vertices to be observed through noisy channels (side information) We prove BP uniqueness holds with and without side information when $d\lambda^2 \ge 1 + C \max\{\lambda, q^{-1}\}\log q$ for some absolute constant $C>0$ independent of $q,d,\lambda$. For large $q$ and $\lambda = o(1/\log q)$, this is asymptotically achieving the Kesten-Stigum threshold $d\lambda^2=1$. These results imply mutual information formula and optimal recovery algorithms for the $q$-community SBM in the corresponding ranges. For $q\ge 4$, Sly (2011); Mossel et al. (2022) shows that there exist choices of $q,d,\lambda$ below Kesten-Stigum (i.e. $d\lambda^2 < 1$) but reconstruction is possible. Somewhat surprisingly, we show that in such regimes BP uniqueness \textit{does not hold} at least in the presence of weak side information. Our technical tool is a theory of q-ary symmetric channels, that we initiate here, generalizing the classical and widely-utilized information-theoretic characterization of BMS (binary memoryless symmetric) channels.
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It is well known, that Fr\'echet means on non-Euclidean spaces may exhibit nonstandard asymptotic rates depending on curvature. Even for distributions featuring standard asymptotic rates, there are non-Euclidean effects, altering finite sampling rates up to considerable sample sizes. These effects can be measured by the variance modulation function proposed by Pennec (2019). Among others, in view of statistical inference, it is important to bound this function on intervals of sampling sizes. In a first step into this direction, for the special case of a K-spider we give such an interval, based only on folded moments and total probabilities of spider legs and illustrate the method by simulations.
In this paper we study multi-task oriented communication system via studying analog encoding method for multiple estimation tasks. The basic idea is to utilize the correlation among interested information required by different tasks and the feature of broadcast channel. For linear estimation tasks, we provide a low complexity design for multi-user multi-task system based on orthogonal decomposition of subspaces. It is proved to be optimal in some special cases, and for general cases, numerical results also show it can achieve near-optimal performance. Further, we make a trial to migrate above method to neural networks based non-linear estimation tasks, and it also shows improvement in energy efficiency.
In this paper we show a polar coding scheme for the deletion channel with a probability of error that decays roughly like $2^{-\sqrt{\Lambda}}$, where $\Lambda$ is the length of the codeword. That is, the same decay rate as that of seminal polar codes for memoryless channels. This is stronger than prior art in which the square root is replaced by a cube root. Our coding scheme is similar yet distinct from prior art. The main differences are: 1) Guard-bands are placed in almost all polarization levels; 2) Trellis decoding is applied to the whole received word, and not to segments of it. As before, the scheme is capacity-achieving. The price we pay for this improvement is a higher decoding complexity, which is nonetheless still polynomial, $O(\Lambda^4)$.
This work concerns controlling the false discovery rate (FDR) in networks under communication constraints. We present sample-and-forward, a flexible and communication-efficient version of the Benjamini-Hochberg (BH) procedure for multihop networks with general topologies. Our method evidences that the nodes in a network do not need to communicate p-values to each other to achieve a decent statistical power under the global FDR control constraint. Consider a network with a total of $m$ p-values, our method consists of first sampling the (empirical) CDF of the p-values at each node and then forwarding $\mathcal{O}(\log m)$ bits to its neighbors. Under the same assumptions as for the original BH procedure, our method has both the provable finite-sample FDR control as well as competitive empirical detection power, even with a few samples at each node. We provide an asymptotic analysis of power under a mixture model assumption on the p-values.
Kernel two-sample tests have been widely used for multivariate data in testing equal distribution. However, existing tests based on mapping distributions into a reproducing kernel Hilbert space are mainly targeted at specific alternatives and do not work well for some scenarios when the dimension of the data is moderate to high due to the curse of dimensionality. We propose a new test statistic that makes use of a common pattern under moderate and high dimensions and achieves substantial power improvements over existing kernel two-sample tests for a wide range of alternatives. We also propose alternative testing procedures that maintain high power with low computational cost, offering easy off-the-shelf tools for large datasets. The new approaches are compared to other state-of-the-art tests under various settings and show good performance. The new approaches are illustrated on two applications: The comparison of musks and non-musks using the shape of molecules, and the comparison of taxi trips started from John F.Kennedy airport in consecutive months. All proposed methods are implemented in an R package kerTests.
We present a formal framework for proving the correctness of set implementations backed by binary-search-tree (BST) and linked lists, which are often difficult to prove correct using automation. This is because many concurrent set implementations admit non-local linearization points for their `contains' procedure. We demonstrate this framework by applying it to the Contention-Friendly Binary-Search Tree algorithm of Crain et al. We took care to structure our framework in a way that can be easily translated into input for model-checking tools such as TLA+, with the aim of using a computer to verify bounded versions of claims that we later proved manually. Although this approach does not provide complete proof (i.e., does not constitute full verification), it allows checking the reasonableness of the claims before spending effort constructing a complete proof. This is similar to the test-driven development methodology, that has proven very beneficial in the software engineering community. We used this approach and validated many of the invariants and properties of the Contention-Friendly algorithm using TLA+. It proved beneficial, as it helped us avoid spending time trying to prove incorrect claims. In one example, TLA+ flagged a fundamental error in one of our core definitions. We corrected the definition (and the dependant proofs), based on the problematic scenario TLA+ provided as a counter-example. Finally, we provide a complete, manual, proof of the correctness of the Contention-Friendly algorithm, based on the definitions and proofs of our two-tiered framework.
Objective. Algorithmic differentiation (AD) can be a useful technique to numerically optimize design and algorithmic parameters by, and quantify uncertainties in, computer simulations. However, the effectiveness of AD depends on how "well-linearizable" the software is. In this study, we assess how promising derivative information of a typical proton computed tomography (pCT) scan computer simulation is for the aforementioned applications. Approach. This study is mainly based on numerical experiments, in which we repeatedly evaluate three representative computational steps with perturbed input values. We support our observations with a review of the algorithmic steps and arithmetic operations performed by the software, using debugging techniques. Main results. The model-based iterative reconstruction (MBIR) subprocedure (at the end of the software pipeline) and the Monte Carlo (MC) simulation (at the beginning) were piecewise differentiable. Jumps in the MBIR function arose from the discrete computation of the set of voxels intersected by a proton path. Jumps in the MC function likely arose from changes in the control flow that affect the amount of consumed random numbers. The tracking algorithm solves an inherently non-differentiable problem. Significance. The MC and MBIR codes are ready for the integration of AD, and further research on surrogate models for the tracking subprocedure is necessary.
The paper considers the distribution of a general linear combination of central and non-central chi-square random variables by exploring the branch cut regions that appear in the standard Laplace inversion process. Due to the original interest from the directional statistics, the focus of this paper is on the density function of such distributions and not on their cumulative distribution function. In fact, our results confirm that the latter is a special case of the former. Our approach provides new insight by generating alternative characterizations of the probability density function in terms of a finite number of feasible univariate integrals. In particular, the central cases seem to allow an interesting representation in terms of the branch cuts, while general degrees of freedom and non-centrality can be easily adopted using recursive differentiation. Numerical results confirm that the proposed approach works well while more transparency and therefore easier control in the accuracy is ensured.
This paper generalizes results in noncoherent space-time block code (STBC) design based on quantum error correction (QEC) to new antenna configurations. Previous work proposed QEC-inspired STBCs for antenna geometries where the number of transmit and receive antennas were equal and a power of two. In this work we extend these results by providing QEC-inspired STBCs applicable to all square antenna geometries and some rectangular geometries where the number of receive antennas is greater than the number of transmit antennas. We derive the maximum-likelihood decoding rule for this family of codes for the special case of Rayleigh fading with additive white Gaussian noise. We present Monte Carlo simulations of the performance of the codes in this environment for a three-antenna square geometry and a three-by-six rectangular geometry. We demonstrate competitive performance for these codes with respect to a popular noncoherent differential code.
The dominating NLP paradigm of training a strong neural predictor to perform one task on a specific dataset has led to state-of-the-art performance in a variety of applications (eg. sentiment classification, span-prediction based question answering or machine translation). However, it builds upon the assumption that the data distribution is stationary, ie. that the data is sampled from a fixed distribution both at training and test time. This way of training is inconsistent with how we as humans are able to learn from and operate within a constantly changing stream of information. Moreover, it is ill-adapted to real-world use cases where the data distribution is expected to shift over the course of a model's lifetime. The first goal of this thesis is to characterize the different forms this shift can take in the context of natural language processing, and propose benchmarks and evaluation metrics to measure its effect on current deep learning architectures. We then proceed to take steps to mitigate the effect of distributional shift on NLP models. To this end, we develop methods based on parametric reformulations of the distributionally robust optimization framework. Empirically, we demonstrate that these approaches yield more robust models as demonstrated on a selection of realistic problems. In the third and final part of this thesis, we explore ways of efficiently adapting existing models to new domains or tasks. Our contribution to this topic takes inspiration from information geometry to derive a new gradient update rule which alleviate catastrophic forgetting issues during adaptation.
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