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We present a novel stochastic variational Gaussian process ($\mathcal{GP}$) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for $\mathcal{GP}$s (CVTGP) is defined in terms of the $\mathcal{GP}$ prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent $\mathcal{GP}$ coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to $\mathcal{O}(M)$, enjoys numerical stability, and maintains $\mathcal{O}(M^3)$ time- and $\mathcal{O}(M^2)$ space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic $\mathcal{GP}$ inference methods.

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

A new $H(\textrm{divdiv})$-conforming finite element is presented, which avoids the need for super-smoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and $C^0$ discontinuous Galerkin methods for the biharmonic equation are derived.

The Ising model is defined by an objective function using a quadratic formula of qubit variables. The problem of an Ising model aims to determine the qubit values of the variables that minimize the objective function, and many optimization problems can be reduced to this problem. In this paper, we focus on optimization problems related to permutations, where the goal is to find the optimal permutation out of the $n!$ possible permutations of $n$ elements. To represent these problems as Ising models, a commonly employed approach is to use a kernel that utilizes one-hot encoding to find any one of the $n!$ permutations as the optimal solution. However, this kernel contains a large number of quadratic terms and high absolute coefficient values. The main contribution of this paper is the introduction of a novel permutation encoding technique called dual-matrix domain-wall, which significantly reduces the number of quadratic terms and the maximum absolute coefficient values in the kernel. Surprisingly, our dual-matrix domain-wall encoding reduces the quadratic term count and maximum absolute coefficient values from $n^3-n^2$ and $2n-4$ to $6n^2-12n+4$ and $2$, respectively. We also demonstrate the applicability of our encoding technique to partial permutations and Quadratic Unconstrained Binary Optimization (QUBO) models. Furthermore, we discuss a family of permutation problems that can be efficiently implemented using Ising/QUBO models with our dual-matrix domain-wall encoding.

We prove that the blocklength $n$ of a linear $3$-query locally correctable code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance $\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb F}|-1)^2}\right)^{1/8}\right)}$. In particular, the blocklength of a linear $3$-query LCC with constant distance over any small field grows exponentially with $k$. This improves on the best prior lower bound of $n \geq \tilde{\Omega}(k^3)$ [AGKM23], which holds even for the weaker setting of $3$-query locally decodable codes (LDCs), and comes close to matching the best-known construction of $3$-query LCCs based on binary Reed-Muller codes, which achieve $n \leq 2^{O(k^{1/2})}$. Because there is a $3$-query LDC with a strictly subexponential blocklength [Yek08, Efr09], as a corollary we obtain the first strong separation between $q$-query LCCs and LDCs for any constant $q \geq 3$. Our proof is based on a new upgrade of the method of spectral refutations via Kikuchi matrices developed in recent works [GKM22, HKM23, AGKM23] that reduces establishing (non-)existence of combinatorial objects to proving unsatisfiability of associated XOR instances. Our key conceptual idea is to apply this method with XOR instances obtained via long-chain derivations, a structured variant of low-width resolution for XOR formulas from proof complexity [Gri01, Sch08].

The polynomial identity lemma (also called the "Schwartz-Zippel lemma") states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is an explicit hitting set for all $n$-variate degree $s$, size $s$ algebraic circuits of size $(s+1)^n$. In this paper, we prove the following results: - Let $\varepsilon > 0$ be a constant. For a sufficiently large constant $n$ and all $s > n$, if we have an explicit hitting set of size $(s+1)^{n-\varepsilon}$ for the class of $n$-variate degree $s$ polynomials that are computable by algebraic circuits of size $s$, then for all $s$, we have an explicit hitting set of size $s^{\exp \circ \exp (O(\log^\ast s))}$ for $s$-variate circuits of degree $s$ and size $s$. That is, if we can obtain a barely non-trivial exponent compared to the trivial $(s+1)^{n}$ sized hitting set even for constant variate circuits, we can get an almost complete derandomization of PIT. - The above result holds when "circuits" are replaced by "formulas" or "algebraic branching programs". This extends a recent surprising result of Agrawal, Ghosh and Saxena (STOC 2018,PNAS 2019) who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most $(s^{n^{0.5 - \delta}})$ (where $\delta>0$ is any constant). Hence, our work significantly weakens the hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial saving over the trivial hitting set, and also presents the first such result for algebraic branching programs and formulas.

The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.

The $\textit{data market design}$ problem is a problem in economic theory to find a set of signaling schemes (statistical experiments) to maximize expected revenue to the information seller, where each experiment reveals some of the information known to a seller and has a corresponding price [Bergemann et al., 2018]. Each buyer has their own decision to make in a world environment, and their subjective expected value for the information associated with a particular experiment comes from the improvement in this decision and depends on their prior and value for different outcomes. In a setting with multiple buyers, a buyer's expected value for an experiment may also depend on the information sold to others [Bonatti et al., 2022]. We introduce the application of deep learning for the design of revenue-optimal data markets, looking to expand the frontiers of what can be understood and achieved. Relative to earlier work on deep learning for auction design [D\"utting et al., 2023], we must learn signaling schemes rather than allocation rules and handle $\textit{obedience constraints}$ $-$ these arising from modeling the downstream actions of buyers $-$ in addition to incentive constraints on bids. Our experiments demonstrate that this new deep learning framework can almost precisely replicate all known solutions from theory, expand to more complex settings, and be used to establish the optimality of new designs for data markets and make conjectures in regard to the structure of optimal designs.

We consider the classical Shiryaev--Roberts martingale diffusion, $(R_t)_{t\ge0}$, restricted to the interval $[0,A]$, where $A>0$ is a preset absorbing boundary. We take yet another look at the well-known phenomenon of quasi-stationarity (time-invariant probabilistic behavior, conditional on no absorbtion hitherto) exhibited by the diffusion in the temporal limit, as $t\to+\infty$, for each $A>0$. We obtain new upper- and lower-bounds for the quasi-stationary distribution's probability density function (pdf), $q_{A}(x)$; the bounds vary in the trade-off between simplicity and tightness. The bounds imply directly the expected result that $q_{A}(x)$ converges to the pdf, $h(x)$, of the diffusion's stationary distribution, as $A\to+\infty$; the convergence is pointwise, for all $x\ge0$. The bounds also yield an explicit upperbound for the gap between $q_{A}(x)$ and $h(x)$ for a fixed $x$. By virtue of integration the bounds for the pdf $q_{A}(x)$ translate into new bounds for the corresponding cumulative distribution function (cdf), $Q_{A}(x)$. All of our results are established explicitly, using certain latest monotonicity properties of the modified Bessel $K$ function involved in the exact closed-form formula for $q_{A}(x)$ recently obtained by Polunchenko (2017). We conclude with a discussion of potential applications of our results in quickest change-point detection: our bounds allow for a very accurate performance analysis of the so-called randomized Shiryaev--Roberts--Pollak change-point detection procedure.

Due to their inherent capability in semantic alignment of aspects and their context words, attention mechanism and Convolutional Neural Networks (CNNs) are widely applied for aspect-based sentiment classification. However, these models lack a mechanism to account for relevant syntactical constraints and long-range word dependencies, and hence may mistakenly recognize syntactically irrelevant contextual words as clues for judging aspect sentiment. To tackle this problem, we propose to build a Graph Convolutional Network (GCN) over the dependency tree of a sentence to exploit syntactical information and word dependencies. Based on it, a novel aspect-specific sentiment classification framework is raised. Experiments on three benchmarking collections illustrate that our proposed model has comparable effectiveness to a range of state-of-the-art models, and further demonstrate that both syntactical information and long-range word dependencies are properly captured by the graph convolution structure.