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Given $n$ independent random vectors with common density $f$ on $\mathbb{R}^d$, we study the weak convergence of three empirical-measure based estimators of the convex $\lambda$-level set $L_\lambda$ of $f$, namely the excess mass set, the…
Inference efficiency in Large Language Models (LLMs) is fundamentally limited by their serial, autoregressive generation, especially as reasoning becomes a key capability and response sequences grow longer. Speculative decoding (SD) offers…
We present and discuss general techniques for proving inapproximability results for truthful mechanisms. We make use of these techniques to prove lower bounds on the approximability of several non-utilitarian multi-parameter problems. In…
Recent studies on frequent itemset mining algorithms resulted in significant performance improvements. However, if the minimal support threshold is set too low, or the data is highly correlated, the number of frequent itemsets itself can be…
We propose a novel approach to conformal prediction for generative language models (LMs). Standard conformal prediction produces prediction sets -- in place of single predictions -- that have rigorous, statistical performance guarantees. LM…
Standard evaluation protocols reveal a counterintuitive phenomenon: on 7.7% of benchmark problems spanning five datasets, larger language models underperform smaller ones by 28.4 percentage points despite 10-100x more parameters. Through…
Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a…
The randomized version of the Kaczmarz method for the solution of linear systems is known to converge linearly in expectation. In this work we extend this result and show that the recently proposed Randomized Sparse Kaczmarz method for…
Regular expressions (res), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic…
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is…
Large language models (LLMs) are often queried multiple times at test time, with predictions aggregated by majority vote. While effective, this self-consistency strategy (arXiv:2203.11171) requires a fixed number of calls and can fail when…
Folklore in complexity theory suspects that circuit lower bounds against $\mathbf{NC}^1$ or $\mathbf{P}/\operatorname{poly}$, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like…
Hardness amplification is a central problem in the study of interactive protocols. While ``natural'' parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message…
We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we…
This article introduces probabilistic disjunctive normal forms (PDNFs) as a framework for representing and reasoning about uncertainty in logical systems. Unlike classical DNFs, PDNFs assign real-valued weights to variables, encoding…
Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of $d \times d$ matrices over a field $\mathbb{F}$, is a…
The AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code is defined as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code.…
In this report, we study partial quantifier elimination (PQE) for propositional CNF formulas. PQE is a generalization of quantifier elimination where one can limit the set of clauses taken out of the scope of quantifiers to a small subset…
Here, in a series of articles, we show methods for calculating propositional statements using algebraic polynomials as symbols for the connectives, which are named operators. These polynomials originate from the transformation between the…
We prove a \emph{query complexity} lower bound on rank-one principal component analysis (PCA). We consider an oracle model where, given a symmetric matrix $M \in \mathbb{R}^{d \times d}$, an algorithm is allowed to make $T$ \emph{exact}…