相关论文: Small Spans in Scaled Dimension

In this paper, we introduce principal asymmetric least squares (PALS) as a unified framework for linear and nonlinear sufficient dimension reduction. Classical methods such as sliced inverse regression (Li, 1991) and principal support…

Large language models (LLMs) have recently garnered significant interest. With in-context learning, LLMs achieve impressive results in various natural language tasks. However, the application of LLMs to sentence embeddings remains an area…

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix…

Let $\Psi$ be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system $\Psi$: $$\sum_{x\in [-N,N]^d} \prod_{i=1}^t…

For many quantum systems of interest, the classical computational cost of simulating their time evolution scales exponentially in the system size. At the same time, quantum computers have been shown to allow for simulations of some of these…

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural…

Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated…

Pseudo-Boolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision…

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semi-algebraic proof systems, the classical constructions of…

Shifted partial derivative (SPD) methods are a central algebraic tool for circuit lower bounds, measuring the dimension of spaces of shifted derivatives of a polynomial. We develop the Shifted Partial Derivative Polynomial (SPDP) framework,…

Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of such scaling law,…

We analyze Kumar's recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases. The lower bound relies on…

Large Language Models (LLMs) have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5)…

We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdos, Lovasz, Nesetril, Rodl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its…

Scaling laws aim to accurately predict model performance across different scales. Existing scaling-law studies almost exclusively rely on cross-entropy as the evaluation metric. However, cross-entropy provides only a partial view of…

The scientific scale-up of large language models (LLMs) necessitates a comprehensive understanding of their scaling properties. However, the existing literature on the scaling properties only yields an incomplete answer: optimization loss…

Lutz (1987) introduced resource-bounded category and showed the circuit size class SIZE($\frac{2^n}{n}$) is meager within ESPACE. Li (2024) established that the symmetric alternation class $S^E_2$ contains problems requiring circuits of…

We develop an extension of recently developed methods for obtaining time-space tradeoff lower bounds for problems of learning from random test samples to handle the situation where the space of tests is signficantly smaller than the space…

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems…

Automated scoring of student work at scale requires balancing accuracy against cost and latency. In "cascade" systems, small language models (LMs) handle easier scoring tasks while escalating harder ones to larger LMs -- but the challenge…