Related papers: Counterexamples to the Low-Degree Conjecture
In recent years much effort was put into developing polynomial-time conditional lower bounds for algorithms and data structures in both static and dynamic settings. Along these lines we suggest a framework for proving conditional lower…
The purpose of this paper is the formal verification of a counterexample of Santos et al. to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra). In contrast with the pen-and-paper proof, our approach is…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
Hypothesis testing in high dimensional data is a notoriously difficult problem without direct access to competing models' likelihood functions. This paper argues that statistical divergences can be used to quantify the difference between…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph…
We comment on some conceptual and and technical problems related to computational mechanics, point out some errors in several papers, and straighten out some wrong priority claims. We present explicitly the correct algorithm for…
A decade ago, a beautiful paper by Wagner developed a ``toolkit'' that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During…
We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial…
Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…
In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low degree polynomial…
The Curry-Howard correspondence is often called the proofs-as-programs result. I offer a generalization of this result, something which may be called machines as programs. Utilizing this insight, I introduce two new Turing Machines called…
We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable…
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of…
In particle physics and cosmology, distinguishing subtle new physics signals from established backgrounds is a fundamental and persistent challenge for phenomenologists. This paper discuss a simple and robust statistical framework to…
Robust classification algorithms have been developed in recent years with great success. We take advantage of this development and recast the classical two-sample test problem in the framework of classification. Based on the estimates of…
We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is…