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We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small…
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties.…
This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning. Recently there has been some research activity in replacing…
We study uniformity conditions for parameterized Boolean circuit families. Uniformity conditions require that the infinitely many circuits in a circuit family are in some sense easy to construct from one shared description. For shallow…
We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989),…
We investigate circuit complexity to characterize chaos in multiparticle quantum systems. In the process, we take a stride to analyze open quantum systems by using complexity. We propose a new diagnostic of quantum chaos from complexity…
This article is devoted to propose some lower and upper bounds for the coupled-tasks scheduling problem in presence of compatibility constraints according to classical complexity hypothesis ($\mathcal{P} \neq \mathcal{NP}$,…
Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can…
The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
Probabilistic circuits (PCs) are a powerful modeling framework for representing tractable probability distributions over combinatorial spaces. In machine learning and probabilistic programming, one is often interested in understanding…
Probabilistic Circuits (PCs) are prominent tractable probabilistic models, allowing for a range of exact inferences. This paper focuses on the main algorithm for training PCs, LearnSPN, a gold standard due to its efficiency, performance,…
A common lens to theoretically study neural net architectures is to analyze the functions they can approximate. However, constructions from approximation theory may be unrealistic and therefore less meaningful. For example, a common…
This document collects the lecture notes from my course "Communication Complexity (for Algorithm Designers),'' taught at Stanford in the winter quarter of 2015. The two primary goals of the course are: 1. Learn several canonical problems…
Complexity classes such as $\#\mathbf{P}$, $\oplus\mathbf{P}$, $\mathbf{GapP}$, $\mathbf{OptP}$, $\mathbf{NPMV}$, or the class of fuzzy languages realised by polynomial-time fuzzy nondeterministic Turing machines, can all be described in…
In this monograph, we study complexity classes that are defined using $O(\log n)$-space bounded non-deterministic Turing machines. We prove salient results of Computational Complexity in this topic such as the Immerman-Szelepcsenyi Theorem,…
Quantum circuit complexity is a fundamental concept whose importance permeates quantum information, computation, many-body physics and high-energy physics. While extensively studied in closed systems, its characterization and behaviors in…
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in…
Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be…
Boolean matching is an important problem in logic synthesis and verification. Despite being well-studied for conventional Boolean circuits, its treatment for reversible logic circuits remains largely, if not completely, missing. This work…