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Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with $n$ nodes and $m$ edges, outputs a classical description of an $\epsilon$-spectral sparsifier in sublinear time $\tilde{O}(\sqrt{mn}/\epsilon)$. This contrasts with the optimal classical complexity $\tilde{O}(m)$. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for $k$-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.

In this paper, we study the computation of the rate-distortion-perception function (RDPF) for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint that belongs to the family of f-divergences. For that, we leverage the fact that RDPF, assuming mild regularity conditions on the perception constraint, forms a convex programming problem. We first develop parametric characterizations of the optimal solution and utilize them in an alternating minimization approach for which we prove convergence guarantees. The resulting structure of the iterations of the alternating minimization approach renders the implementation of a generalized Blahut-Arimoto (BA) type of algorithm infeasible. To overcome this difficulty, we propose a relaxed formulation of the structure of the iterations in the alternating minimization approach, which allows for the implementation of an approximate iterative scheme. This approximation is shown, via the derivation of necessary and sufficient conditions, to guarantee convergence to a globally optimal solution. We also provide sufficient conditions on the distortion and the perception constraints which guarantee that our algorithm converges exponentially fast. We corroborate our theoretical results with numerical simulations, and we draw connections with existing results.

Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank sub-determinants of $A$ are bounded by $\Delta$ in the absolute value. We present a new FPT-algorithm, parameterized by $\Delta$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $\Delta$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $\Delta$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $\Delta$-modular subset-sum problem.

We show that the sparsified block elimination algorithm for solving undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman STOC'16] directly works for directed Laplacians. Given access to a sparsification algorithm that, on graphs with $n$ vertices and $m$ edges, takes time $\mathcal{T}_{\rm S}(m)$ to output a sparsifier with $\mathcal{N}_{\rm S}(n)$ edges, our algorithm solves a directed Eulerian system on $n$ vertices and $m$ edges to $\epsilon$ relative accuracy in time $$ O(\mathcal{T}_{\rm S}(m) + {\mathcal{N}_{\rm S}(n)\log {n}\log(n/\epsilon)}) + \tilde{O}(\mathcal{T}_{\rm S}(\mathcal{N}_{\rm S}(n)) \log n), $$ where the $\tilde{O}(\cdot)$ notation hides $\log\log(n)$ factors. By previous results, this implies improved runtimes for linear systems in strongly connected directed graphs, PageRank matrices, and asymmetric M-matrices. When combined with slower constructions of smaller Eulerian sparsifiers based on short cycle decompositions, it also gives a solver that runs in $O(n \log^{5}n \log(n / \epsilon))$ time after $O(n^2 \log^{O(1)} n)$ pre-processing. At the core of our analyses are constructions of augmented matrices whose Schur complements encode error matrices.

A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute a weak distance between them using the Fourier transform on the ambient space. In particular, the inhomogeneous Sobolev norm of negative order for a difference between two surface measures is evaluated via the Plancherel theorem, which amounts to approximating an weighted integral norm of smooth data on the frequency space. This approach allows several advantages including high accuracy due to fast-converging numerical quadrature rules, acceleration by the nonuniform fast Fourier transform, and parallelization on many-core processors. In numerical experiments, the 2-sphere, which is an example whose Fourier transform is explicitly known, is compared with its icosahedral discretization, and it is observed that the piecewise linear approximations converge to the smooth object at the quadratic rate up to small truncation.

Diffusion models have emerged as a key pillar of foundation models in visual domains. One of their critical applications is to universally solve different downstream inverse tasks via a single diffusion prior without re-training for each task. Most inverse tasks can be formulated as inferring a posterior distribution over data (e.g., a full image) given a measurement (e.g., a masked image). This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. To cope with this challenge, we propose a variational approach that by design seeks to approximate the true posterior distribution. We show that our approach naturally leads to regularization by denoising diffusion process (RED-Diff) where denoisers at different timesteps concurrently impose different structural constraints over the image. To gauge the contribution of denoisers from different timesteps, we propose a weighting mechanism based on signal-to-noise-ratio (SNR). Our approach provides a new variational perspective for solving inverse problems with diffusion models, allowing us to formulate sampling as stochastic optimization, where one can simply apply off-the-shelf solvers with lightweight iterates. Our experiments for image restoration tasks such as inpainting and superresolution demonstrate the strengths of our method compared with state-of-the-art sampling-based diffusion models.

This paper explores the capacity of additive Vertically-Drifted First Arrival Position (VDFAP) noise channels, which are emerging as a new paradigm for diffusive molecular communication. Analogous to the capacity of parallel Gaussian channels, the capacity of VDFAP noise channels is defined as the supremum of the mutual information between the input and output signals subject to an overall second-moment constraint on input distributions. Upper and lower bounds for this capacity are derived for the case of three spatial dimensions, based on an analysis of the characteristic function of the VDFAP distribution and an investigation of its stability properties. The results of this study contribute to the ongoing effort to understand the fundamental limits of molecular communication systems.

Beame et al. [ITCS 2018 & TALG 2021] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short span of time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell, Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 & Theory Comput. Syst. 2021], and Chen et al. [SODA 2020]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.'s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries. In this work, we completely resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty and Khanna [ESA 2021] to test graph connectivity.

In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n^3 log^3 n), the first progress since McShine and Tetali's O(n^5 log n) bound in 1997. In the process we give lower and upper bounds of respectively Omega(1/(sqrt n log n)) and O(1/sqrt n) -- asymptotically tight up to an O(log n) factor -- for the expansion of the associahedron graph K_n. The upper bound recovers Molloy, Reed, and Steiger's Omega(n^{3/2}) bound on the mixing time of the walk. To obtain these results, we introduce a framework consisting of a set of sufficient conditions under which a given Markov chain mixes rapidly. This framework is a purely combinatorial analogue that in some circumstances gives better results than the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda. In particular, in addition to the result for triangulations, we show quasipolynomial mixing for the k-angulation flip walk on a convex point set, for fixed k >= 4.