Related papers: Compression is all you need: Modeling Mathematics
Mathematical reasoning and optimization are fundamental to artificial intelligence and computational problem-solving. Recent advancements in Large Language Models (LLMs) have significantly improved AI-driven mathematical reasoning, theorem…
Mathematical morphology contributes many profitable tools to image processing area. Some of these methods considered to be basic but the most important fundamental of data processing in many various applications. In this paper, we modify…
This paper develops multihead finite-state compression, a generalization of finite-state compression, complementary to the multihead finite-state dimensions of Huang, Li, Lutz, and Lutz (2025). In this model, an infinite sequence of symbols…
We give an algorithm that learns a representation of data through compression. The algorithm 1) predicts bits sequentially from those previously seen and 2) has a structure and a number of computations similar to an autoencoder. The…
Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
Classical mathematical models used in the semantics of programming languages and computation rely on idealized abstractions such as infinite-precision real numbers, unbounded sets, and unrestricted computation. In contrast, concrete…
Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such…
For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to $\Sigma^*$? And for which sets A does there exist such a…
Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous…
Despite the increasing prevalence of large language models (LLMs), we still have a limited understanding of how their representational spaces are structured. This limits our ability to interpret how and what they learn or relate them to…
Binary embedding is a nonlinear dimension reduction methodology where high dimensional data are embedded into the Hamming cube while preserving the structure of the original space. Specifically, for an arbitrary $N$ distinct points in…
Learning, prediction, and compression are intimately connected: a model that accurately predicts the next symbol in a sequence can be coupled with a source coder to compress that sequence near its information-theoretic limit. When tokenized…
Compression and generalization are fundamentally related through Solomonoff induction and the minimum description length principle (MDL), which predict that simpler models generalize better when data arises from low-complexity…
Explicit high-order feature interactions efficiently capture essential structural knowledge about the data of interest and have been used for constructing generative models. We present a supervised discriminative High-Order Parametric…
We study the problem of representing all distances between $n$ points in $\mathbb R^d$, with arbitrarily small distortion, using as few bits as possible. We give asymptotically tight bounds for this problem, for Euclidean metrics, for…
Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices. The concept of model compression is also important for…
Sampling-based decoding underlies complex reasoning in large language models (LLMs), where decoding strategies critically shape model behavior. Temperature- and truncation-based methods reshape the next-token distribution through global…
'Skolem arithmetic' is the complete theory $T$ of the multiplicative monoid $(\mathbb{N},\cdot)$. We give a full characterization of the $\varnothing$-definable stably embedded sets of $T$, showing in particular that, up to the relation of…
Manifold learning (ML), known also as non-linear dimension reduction, is a set of methods to find the low dimensional structure of data. Dimension reduction for large, high dimensional data is not merely a way to reduce the data; the new…