The fundamental problem in much of physics and quantum chemistry is to optimize a low-degree polynomial in certain anticommuting variables. Being a quantum mechanical problem, in many cases we do not know an efficient classical witness to the optimum, or even to an approximation of the optimum. One prominent exception is when the optimum is described by a so-called "Gaussian state", also called a free fermion state. In this work we are interested in the complexity of this optimization problem when no good Gaussian state exists. Our primary testbed is the Sachdev--Ye--Kitaev (SYK) model of random degree-$q$ polynomials, a model of great current interest in condensed matter physics and string theory, and one which has remarkable properties from a computational complexity standpoint. Among other results, we give an efficient classical certification algorithm for upper-bounding the largest eigenvalue in the $q=4$ SYK model, and an efficient quantum certification algorithm for lower-bounding this largest eigenvalue; both algorithms achieve constant-factor approximations with high probability.
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We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'{e} approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Pad\'{e} problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.
In a cooperative multiagent system, a collection of agents executes a joint policy in order to achieve some common objective. The successful deployment of such systems hinges on the availability of reliable inter-agent communication. However, many sources of potential disruption to communication exist in practice, such as radio interference, hardware failure, and adversarial attacks. In this work, we develop joint policies for cooperative multiagent systems that are robust to potential losses in communication. More specifically, we develop joint policies for cooperative Markov games with reach-avoid objectives. First, we propose an algorithm for the decentralized execution of joint policies during periods of communication loss. Next, we use the total correlation of the state-action process induced by a joint policy as a measure of the intrinsic dependencies between the agents. We then use this measure to lower-bound the performance of a joint policy when communication is lost. Finally, we present an algorithm that maximizes a proxy to this lower bound in order to synthesize minimum-dependency joint policies that are robust to communication loss. Numerical experiments show that the proposed minimum-dependency policies require minimal coordination between the agents while incurring little to no loss in performance; the total correlation value of the synthesized policy is one fifth of the total correlation value of the baseline policy which does not take potential communication losses into account. As a result, the performance of the minimum-dependency policies remains consistently high regardless of whether or not communication is available. By contrast, the performance of the baseline policy decreases by twenty percent when communication is lost.
We introduce two new metrics of "simplicity" for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with $9.25n+O(1)$ turns and $12n+O(1)$ crossings on an $n\times n$ board, and we show lower bounds of $(6-\epsilon)n$ and $4n-O(1)$ on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of $9.25/6+o(1)$ and $3+o(1)$. Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in $O(n^2/p)$ time using $p$ processors in the CREW PRAM model. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for $(1,4)$-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.
We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some H\"older space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $\frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $\frac{\log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates $\{\frac{1}{T},\frac{\log T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
Most of the popular dependence measures for two random variables $X$ and $Y$ (such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's $\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to $1$ imply that one variable is a measurable function of the other. Yet, both properties are natural desiderata for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman's $\rho$ and a rearranged Kendall's $\tau$ which do attain the value $1$ if, and only if, one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study.
Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms
Motivated by applications in single-cell biology and metagenomics, we consider matrix reordering based on the noisy disordered matrix model. We first establish the fundamental statistical limit for the matrix reordering problem in a decision-theoretic framework and show that a constrained least square estimator is rate-optimal. Given the computational hardness of the optimal procedure, we analyze a popular polynomial-time algorithm, spectral seriation, and show that it is suboptimal. We then propose a novel polynomial-time adaptive sorting algorithm with guaranteed improvement on the performance. The superiority of the adaptive sorting algorithm over the existing methods is demonstrated in simulation studies and in the analysis of two real single-cell RNA sequencing datasets.
Integrated sensing and communication (ISAC) has opened up numerous game-changing opportunities for realizing future wireless systems. In this paper, we propose an ISAC processing framework relying on millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) systems. Specifically, we provide a compressed sampling (CS) perspective to facilitate ISAC processing, which can not only recover the large-scale channel state information or/and radar imaging information, but also significantly reduce pilot overhead. First, an energy-efficient widely spaced array (WSA) architecture is tailored for the radar receiver, which enhances the angular resolution of radar sensing at the cost of angular ambiguity. Then, we propose an ISAC frame structure for time-variant ISAC systems considering different timescales. The pilot waveforms are judiciously designed by taking into account both CS theories and hardware constraints. Next, we design the dedicated dictionary for WSA that serves as a building block for formulating the ISAC processing as sparse signal recovery problems. The orthogonal matching pursuit with support refinement (OMP-SR) algorithm is proposed to effectively solve the problems in the existence of the angular ambiguity. We also provide a framework for estimating and compensating the Doppler frequencies during payload data transmission to guarantee communication performances. Simulation results demonstrate the good performances of both communications and radar sensing under the proposed ISAC framework.
Reasoning about uncertainty is vital in many real-life autonomous systems. However, current state-of-the-art planning algorithms cannot either reason about uncertainty explicitly, or do so with a high computational burden. Here, we focus on making informed decisions efficiently, using reward functions that explicitly deal with uncertainty. We formulate an approximation, namely an abstract observation model, that uses an aggregation scheme to alleviate computational costs. We derive bounds on the expected information-theoretic reward function and, as a consequence, on the value function. We then propose a method to refine aggregation to achieve identical action selection with a fraction of the computational time.
In this paper, we study the \emph{type graph}, namely a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length $n$ of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of $\left(R_{1},R_{2}\right)$ such that there exists a biclique of the type graph which has respectively $e^{nR_{1}}$ and $e^{nR_{2}}$ vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then apply our results and proof ideas to noninteractive simulation problems. We completely characterize the exponents of maximum and minimum joint probabilities when the marginal probabilities vanish exponentially fast with given exponents. These results can be seen as strong small-set expansion theorems. We extend the noninteractive simulation problem by replacing Boolean functions with arbitrary nonnegative functions, and obtain new hypercontractivity inequalities which are stronger than the common hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types, linear algebra, and coupling techniques.
Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
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