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Third-order statistics of the cosmic density field provides a powerful cosmological probe containing synergistic information to the more commonly explored second-order statistics. Here, we exploit a spectroscopic catalog of 72,563 clusters…
In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to…
Quantum Measure Theory (QMT) is a generalization of quantum theory where physical predictions are computed from a matrix known as \emph{decoherence functional} (DF). Previous works have noted that, in its original formulation, QMT exhibits…
Safety and stability are essential properties of control systems. Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) are powerful tools to ensure safety and stability respectively. However, previous approaches typically…
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…
This paper describes a purely functional library for computing level-$p$-complexity of Boolean functions, and applies it to two-level iterated majority. Boolean functions are simply functions from $n$ bits to one bit, and they can describe…
In this paper, we present a construction of linear codes over $\f_{2^t}$ from Boolean functions, which is a generalization of Ding's method \cite[Theorem 9]{Ding15}. Based on this construction, we give two classes of linear codes…
We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a…
This work considers a problem of estimating a mixing probability density $f$ in the setting of discrete mixture models. The paper consists of three parts. The first part focuses on the construction of an $L_1$ consistent estimator of $f$.…
Binary Decision Diagrams (BDDs) are widely used for the representation of Boolean functions. Context-Free-Language Ordered Decision Diagrams (CFLOBDDs) are a plug-compatible replacement for BDDs -- roughly, they are BDDs augmented with a…
Composition technologies improve reuse in the development of large-scale complex systems. Safety critical systems require intensive validation and verification activities. These activities should be compositional in order to reduce the…
Compositional generalization--understanding unseen combinations of seen primitives--is an essential reasoning capability in human intelligence. The AI community mainly studies this capability by fine-tuning neural networks on lots of…
Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of…
State-of-the-art machine learning methods exhibit limited compositional generalization. At the same time, there is a lack of realistic benchmarks that comprehensively measure this ability, which makes it challenging to find and evaluate…
The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$…
A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential…
For dynamical systems with infinite topological entropy, the classical entropy fails to quantify their complexity effectively, while the metric mean dimension provides a natural extension in this context. In this paper, we study the…
Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is…
The complexity of a well-quasi-order (wqo) can be measured through three ordinal invariants: the width as a measure of antichains, height as a measure of chains, and maximal order type as a measure of bad sequences. We study these ordinal…
Safety-critical control is imperative for deploying autonomous systems in the real world. Control Barrier Functions (CBFs) offer strong safety guarantees when accurate system and sensor models are available. However, widely used additive,…