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Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is…
Recent research reveals that machine learning (ML) models are highly sensitive to minor changes in their training procedure, such as the inclusion or exclusion of a single data point, leading to conflicting predictions on individual data…
Satisfiability is a classic problem in computational complexity theory, in which one wishes to determine whether an assignment of values to a collection of Boolean variables exists in which all of a collection of clauses composed of logical…
The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid…
The satisfiability threshold for constraint satisfaction problems is that value of the ratio of constraints (or clauses) to variables, above which the probability that a random instance of the problem has a solution is zero in the large…
Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among…
In this paper we revisit random linear under-determined systems with sparse solutions. We consider $\ell_1$ optimization heuristic known to work very well when used to solve these systems. A collection of fundamental results that relate to…
I present an analytic approach to establishing the presence of phase transitions in a large set of decision problems. This approach does not require extensive computational study of the problems considered. The set -- that of all paddable…
We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of 3-loops (triangles). We find that such ensembles generally exhibit two transitions, a clustering…
Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the $\ell_1$ minimization method for…
In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual…
Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and…
The random K-satisfiability (K-SAT) problem is an important problem for studying typical-case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity spin-glasses. In this paper we…
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have reveals that this anomalous scaling causes a…
This paper first analyzes the resolution complexity of two random CSP models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it…
In principle, the probability of configurations, determined by the system's partition function or wave function, encapsulates essential information about phases and phase transitions. Despite the exponentially large configuration space, we…
We present results from density functional theory and computer simulations that unambiguously predict the occurrence of first-order freezing transitions for a large class of ultrasoft model systems into cluster crystals. The clusters…
We propose a general variational framework of fair clustering, which integrates an original Kullback-Leibler (KL) fairness term with a large class of clustering objectives, including prototype or graph based. Fundamentally different from…
Transitions between multiple stable states of nonlinear systems are ubiquitous in physics, chemistry, and beyond. Two types of behaviors are usually seen as mutually exclusive: unpredictable noise-induced transitions and predictable…
We analyze the emergence of diffractive focusing in the transition from discrete to continuous space-time variables. Three types of dynamical equations are studied in a top-to-bottom approach, starting with the most general system. First we…