New Fourier analysis method for non-uniform Boolean hypercube.
problem Non-uniform probability measures on the Boolean hypercube.
method ANOVA-based decomposition, explicit basis, least squares problem.
result Generalization of Fourier analysis for arbitrary probability measures.
Paper improves variational inference on Boolean hypercube using quantum methods.
problem Improving variational inference for pairwise Markov random fields on the Boolean hypercube.
method Quantum relaxations of the Kullback-Leibler divergence for upper-bounds, primal-dual optimization, and greedy selection of hierarchies.
result Efficient algorithm and improved bounds for variational inference.
New algorithm learns halfspaces over hypercube with random bit flips.
problem Agnostic learning of Boolean halfspaces over discrete domains is computationally hard.
method Smoothed analysis with random bit flips for discrete inputs.
result First efficient algorithm for smoothed agnostic learning of halfspaces over Boolean hypercube.
This work extends score-based methods to binary data on the Boolean hypercube.
problem Learning and sampling binary data on the Boolean hypercube.
method Adopting Bernoulli noise as a smoothing device, deriving a TMF-like expression for the optimal denoiser, and using a Langevin-like sampler.
result The method successfully samples noisy binary data and reduces effective noise through multiple measurements.
Efficiently optimizes boolean functions using multilinear polynomials and exponential weight updates.
problem Optimizing boolean functions over the boolean hypercube with high computational cost.
method Proposes a computationally efficient algorithm using multilinear polynomials and exponential weight updates.
result Improves computational time up to several orders of magnitude compared to state-of-the-art algorithms.
The staircase property aids deep learning by guiding hierarchical feature learning.
problem Understanding how hierarchical structure influences deep learning performance.
method Defined and proved the staircase property for Boolean hypercube functions, and showed its learnability by layerwise stochastic coordinate descent.
result Staircase functions can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks.
New findings show transfer learning is possible even when density ratios are unbounded.
problem Transfer learning under unbounded density ratios.
method Low-degree polynomial estimators, general transfer inequality over Rn. result Non-trivial transfer learning possible under mild assumptions, including log-concave measures.
We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches t…
The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In t…
We quantify the separation between the numbers of labeled examples required to learn in two settings: Settings with and without the knowledge of the distribution of the unlabeled data. More specifically, we prove a separation by Θ(logn) multiplicative factor for the class of projections over the Boolean hypercube o…
New bounds for learning polynomial surrogates with L∞ guarantees.
problem Learning polynomial surrogates for bounded binary functions with L∞ error guarantees. method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as nd+1 for degree d polynomials and ns2 for sparse polynomials. The hypercube's perimeter is significantly larger than expected near half volume.
problem Understanding the isoperimetric profile of the hypercube.
method Analytical proof of perimeter bounds and comparison to Gaussian isoperimetric profile.
result The isoperimetric profile of the hypercube does not converge to the Gaussian profile as dimension increases.
Boolean logic used for neural network training and inference, with convergence analysis.
problem Discrete optimization in neural networks with Boolean logic.
method Boolean logic backpropagation with convergence analysis.
result First convergence analysis for Boolean logic in neural networks.
Improved agnostic learning time via Gaussian surface area analysis.
problem Learning polynomial threshold functions under Gaussian marginals.
method Improvement of polynomial degree required for approximation.
result Near optimal bounds on agnostic learning complexity.
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…
Study shows SQ lower bounds for learning ReLUs with Massart noise.
problem Learning a single neuron in the presence of Massart noise.
method Statistical Query (SQ) lower bounds for efficient learning algorithms.
result No efficient SQ algorithm can approximate the optimal error within any constant factor.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
problem Spectral rigidity of hypercube graphs
method Interplay between global spectral embedding and local curvature analysis
result Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
A new deep learning method using Boolean logic reduces training and inference energy.
problem High computational and energy costs in deep learning training and inference.
method Introduces Boolean weights and inputs for efficient training using Boolean logic.
result Achieves full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation.
Boolean matrix factorization and Boolean matrix completion from noisy observations are desirable unsupervised data-analysis methods due to their interpretability, but hard to perform due to their NP-hardness. We treat these problems as maximum a posteriori inference problems in a graphical model and present a message p…
New Boolean algebra method shows knot unknotting number is (c+1)/2.
problem Finding the minimum number of region crossing changes to unknot a knot.
method Boolean algebra applied to region crossing changes.
result Region unknotting number is (c+1)/2 for any knot with crossing number c.
A new method relaxes Boolean Matrix Factorization to make it more efficient.
problem High computational cost of solving NP-hard combinatorial optimization problems in Boolean Matrix Factorization.
method Proposes a proximal gradient algorithm using an elastic-binary regularizer to relax BMF.
result Demonstrates improved runtime and better recall, loss, and interpretability on real-world data.
Probabilistic learning for binary classification with categorical variables.
problem Binary classification with categorical covariates.
method Probabilistic analysis and two algorithms for learning boolean functions.
result Effective learning of boolean functions from binary data.
Boolean matrix factorisation aims to decompose a binary data matrix into an approximate Boolean product of two low rank, binary matrices: one containing meaningful patterns, the other quantifying how the observations can be expressed as a combination of these patterns. We introduce the OrMachine, a probabilistic genera…
Survey on learning Boolean functions in computational theory.
problem Learning Boolean function classes in computational theory.
method Overview of known results in PAC and related models.
result Discussion of various learning results for Boolean functions.
Simplified Khovanov polynomials for bipartite links.
problem Computing Khovanov polynomials for bipartite links.
method Reduced Khovanov-Rozansky technique to Kauffman-Khovanov cycle calculus.
result Consistency demonstrated between reduced technique and bipartite Khovanov polynomials.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.
problem Understanding the structure of graphs with specific curvature conditions.
method Analyzing weighted graphs with lower Ricci curvature bounds and eigenvalue closeness to establish structural similarity.
result Discrete graphs with specific curvature conditions are close to hypercube structures in terms of Frobenius distance and eigenfunctions.
It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of robust learning from the perspective of computational learning theory, considering both sample and computational complexity. In particular, our definition of…
A Bayesian Boolean Matrix Factorization for cancer genomics
problem Identifying coordinated feature changes in cancer
method Bayesian Boolean Matrix Factorization
result Captures widespread, near-simultaneous chromosome-number changes
Minimalist softmax attention learns constrained Boolean functions with supervision.
problem Learning k-bit Boolean functions, especially AND and OR. method Minimalist single-head softmax-attention mechanism with teacher forcing.
result Minimalist attention can solve Boolean tasks with supervision, but not without.
New framework reduces LLM complexity by directly finetuning in Boolean domain.
problem Reducing the complexity of large language models (LLMs) while maintaining performance.
method Proposes a novel framework using multi-kernel Boolean parameters for direct finetuning in the Boolean domain.
result Significantly reduces complexity during both finetuning and inference, outperforming recent techniques.
GETF efficiently decomposes large-scale Boolean tensors.
problem Efficiently factorizing large-scale Boolean tensors.
method Geometric Expansion for all-order Tensor Factorization (GETF).
result GETF significantly improves reconstruction accuracy and efficiency.
The paper develops sum-of-squares relaxations for computing f-divergences.
problem Computing f-divergences from non-centered covariance matrices. method Sum-of-squares relaxations for convex optimization.
result Sum-of-squares relaxations make computations tractable.
LIBRE learns interpretable Boolean rules from data.
problem Creating interpretable classifiers in imbalanced settings.
method Ensemble of weak learners on random feature subsets, combined with a simple union.
result Efficiently balances prediction accuracy and interpretability.
Transformers learn sparse Boolean functions through RL and SFT, revealing distinct learning behaviors.
problem Learning sparse Boolean functions with Transformers.
method Reinforcement Learning (RL) with process rewards and Supervised Fine-Tuning (SFT).
result RL learns the whole CoT chain simultaneously, while SFT learns step by step.
Graph neural networks struggle with proving unsatisfiability in complex logical formulas.
problem Proving unsatisfiability in complex logical formulas.
method Investigating the limitations of graph neural networks in logical reasoning tasks.
result Graph neural networks may fail in certifying unsatisfiability in Boolean formulae.
A Boolean algebra formalizes task composition for reinforcement learning.
problem Formalizing task composition for efficient learning and problem-solving.
method Formalized tasks as a Boolean algebra, learning goal-oriented value functions, and composing them to solve new tasks.
result Agents can solve new tasks without additional learning by composing value functions in specific ways.
Paper proposes algorithms for BMF using integer programming.
problem Approximating binary input matrix as product of two smaller binary factors.
method Alternating optimization strategy using integer programming to solve subproblems and combine solutions.
result Proposed algorithms outperform state of the art on medium-scale problems.
The paper develops algorithms for Boolean matrix factorization using IP and heuristics.
problem Approximating binary input matrices as products of smaller binary factors.
method Alternating optimization with integer programming and greedy/local-search heuristics.
result Proposed methods improve scalability and performance compared to existing techniques.
Boolean matrix factorization (BMF) is a popular and powerful technique for inferring knowledge from data. The mining result is the Boolean product of two matrices, approximating the input dataset. The Boolean product is a disjunction of rank-1 binary matrices, each describing a feature-relation, called pattern, for a g…
Study examines noise sensitivity of DNNs for binary classification.
problem Understanding non-robustness of DNN classifiers under noise.
method Defined and extended noise sensitivity and stability concepts for Boolean functions, applied to DNN models.
result Sorted out the relation between definitions and properties of DNN architectures under noise.
This paper explores how boolean formulas can be learned by deep neural networks.
problem Understanding the learnability of boolean formulas by deep neural networks.
method Analysis of boolean formulas associated with model-sampling benchmarks, combinatorial optimization problems, and random 3-CNFs.
result Neural networks outperform rule-based systems and pure symbolic approaches in learning boolean formulas.
We explain why numbers occurring in the classification of polygon spaces coincide with numbers of self-dual equivalence classes of threshold functions, or of regular Boolean functions, or of decisive weighted majority games.
This paper introduces the combinatorial Boolean model (CBM), which is defined as the class of linear combinations of conjunctions of Boolean attributes. This paper addresses the issue of learning CBM from labeled data. CBM is of high knowledge interpretability but naïve learning of it requires exponentially large compu…
Theory of ends of spaces using linear algebra.
problem Understanding ends of spaces at infinity.
method Developing a theory using scale, sub-Boolean algebras, and linear algebra.
result All known types of ends are special cases of a linear algebraic process.
We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of n objects is chosen, and a linear cost associated with the chosen subset is incurred. To measure the performance of our algorithms, we use the notion of regret which is the difference between the to…
Boolean matrix has been used to represent digital information in many fields, including bank transaction, crime records, natural language processing, protein-protein interaction, etc. Boolean matrix factorization (BMF) aims to find an approximation of a binary matrix as the Boolean product of two low rank Boolean matri…