Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques

The tensor power of the clique on t vertices (denoted by K_t^n) is the graph on vertex set {1, ..., t}^n such that two vertices x, y in {1, ..., t}^n are connected if and only if x_i != y_i for all i in {1, ..., n}. Let the density of a subset S of K_t^n to be mu(S) := |S|/t^n. Also let the vertex boundary of a set S to be the vertices of the graph, including those of S, which are incident to some vertex of S. We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density nu in [0, 1] what is the smallest possible density of the vertex boundary of a subset of K_t^n of density nu? Let Phi_t(nu) be the infimum of these minimum densities as n -> infinity. We find a recursive relation allows one to compute Phi_t(nu) in time polynomial to the number of desired bits of precision. Second, we study given an independent set I of K_t^n of density mu(I) = (1-epsilon)/t, how close it is to a maximum-sized independent set J of density 1/t. We show that this deviation (measured by mu(IJ)) is at most 4 epsilon^{(log t)/(log t - log(t-1))} as long as epsilon < 1 - 3/t + 2/t^2. This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an O(epsilon) upper bound. We also show the exponent (log t)/(log t - log(t-1)) is optimal assuming n tending to infinity and epsilon tending to 0. The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings. The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Quantum meets optimization and machine learning

With the advent of the quantum era, what role the quantum computer will play in optimization and machine learning becomes a natural and salient question. The development of novel quantum computing techniques is essential to showcase the quantum advantage in these fields. At the same time, new findings in classical optimization and machine learning algorithms also have the potential to stimulate quantum computing research. In the dissertation, we explore the fascinating connections between quantum computing, optimization, and machine learning, paving the way for transformative advances in all three fields. First, on the quantum side, we present efficient quantum algorithms for fundamental problems in sampling, optimization, and quantum physics. Our results highlight the practical advantages of quantum computing in these fields. In addition, we introduce new approaches to quantum complexity theory for characterizing the quantum hardness of optimization and machine learning problems. Second, on the optimization side, we improve the efficiency of the state-of-the-art classical algorithms for solving semi-definite programming (SDP), Fourier sensing, dynamic distance estimation, etc. Our classical results are closely intertwined with quantum computing. Some of them serve as stepping stones to new quantum algorithms, while others are motivated by quantum applications or inspired by quantum techniques. Third, on the machine learning side, we develop fast classical and quantum algorithms for training over-parameterized neural networks with provable guarantees of convergence and generalization. Furthermore, we contribute to the security aspect of machine learning by theoretically investigating some potential approaches to (classically) protect private data in collaborative machine learning and to (quantumly) protect the copyright of machine learning models. Fourth, we investigate the concentration and discrepancy properties of hyperbolic polynomials and higher-order random walks, which could potentially be applied to quantum computing, optimization, and machine learning.Computer Science