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In this work we have considered formal power series and partial differential equations, and their relationship with Coding Theory. We have obtained the nature of solutions for the partial differential equations for Cycle Poisson Case. The…

Differentiation is a cornerstone of computing and data analysis in every discipline of science and engineering. Indeed, most fundamental physics laws are expressed as relationships between derivatives in space and time. However, derivatives…

In this work we explore deep generative models of text in which the latent representation of a document is itself drawn from a discrete language model distribution. We formulate a variational auto-encoder for inference in this model and…

Lazy evaluation is a powerful tool that enables better compositionality and potentially better performance in functional programming, but it is challenging to analyze its computation cost. Existing works either require manually annotating…

Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…

Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…

A natural model of read-once linear branching programs is a branching program where queries are $\mathbb{F}_2$ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we…

Real-world data often exhibits sequential dependence, across diverse domains such as human behavior, medicine, finance, and climate modeling. Probabilistic methods capture the inherent uncertainty associated with prediction in these…

We describe derivations of several important associative and Lie rings of infinite matrices over general rings of coefficients.

Models characterized by autoregressive structure and random coefficients are powerful tools for the analysis of high-frequency, high-dimensional and volatile time series. The available literature on such models is broad, but also sectorial,…

The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…

Speculative decoding is a technique that uses multiple language models to accelerate infer- ence. Previous works have used an experi- mental approach to optimize the throughput of the inference pipeline, which involves LLM training and can…

We want to obtain derivatives in discontinuous program code, where default Algorithmic Differentiation may not perform well. Specifically, we consider discontinuities induced by control flow statements, where meaningful derivatives should…

Neural codes are collections of binary strings motivated by patterns of neural activity. In this paper, we study algorithmic and enumerative aspects of convex neural codes in dimension 1 (i.e. on a line or a circle). We use the theory of…

Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…

We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. We argue that the task of program learning should be more tractable for these architectures…

Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…

In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have…

We introduce a novel generative formulation of deep probabilistic models implementing "soft" constraints on their function dynamics. In particular, we develop a flexible methodological framework where the modeled functions and derivatives…

State of the art machine learning algorithms are highly optimized to provide the optimal prediction possible, naturally resulting in complex models. While these models often outperform simpler more interpretable models by order of…