Symbolic Calculus in Mathematical Statistics: A Review

In the last ten years, the employment of symbolic methods has substantially extended both the theory and the applications of statistics and probability. This survey reviews the development of a symbolic technique arising from classical umbral calculus, as introduced by Rota and Taylor in $1994.$ The usefulness of this symbolic technique is twofold. The first is to show how new algebraic identities drive in discovering insights among topics apparently very far from each other and related to probability and statistics. One of the main tools is a formal generalization of the convolution of identical probability distributions, which allows us to employ compound Poisson random variables in various topics that are only somewhat interrelated. Having got a different and deeper viewpoint, the second goal is to show how to set up algorithmic processes performing efficiently algebraic calculations. In particular, the challenge of finding these symbolic procedures should lead to a new method, and it poses new problems involving both computational and conceptual issues. Evidence of efficiency in applying this symbolic method will be shown within statistical inference, parameter estimation, L\'evy processes, and, more generally, problems involving multivariate functions. The symbolic representation of Sheffer polynomial sequences allows us to carry out a unifying theory of classical, Boolean and free cumulants. Recent connections within random matrices have extended the applications of the symbolic method.Comment: 72 page

Laughing Hyena Distillery: Extracting Compact Recurrences From Convolutions

Recent advances in attention-free sequence models rely on convolutions as alternatives to the attention operator at the core of Transformers. In particular, long convolution sequence models have achieved state-of-the-art performance in many domains, but incur a significant cost during auto-regressive inference workloads -- naively requiring a full pass (or caching of activations) over the input sequence for each generated token -- similarly to attention-based models. In this paper, we seek to enable $\mathcal O(1)$ compute and memory cost per token in any pre-trained long convolution architecture to reduce memory footprint and increase throughput during generation. Concretely, our methods consist in extracting low-dimensional linear state-space models from each convolution layer, building upon rational interpolation and model-order reduction techniques. We further introduce architectural improvements to convolution-based layers such as Hyena: by weight-tying the filters across channels into heads, we achieve higher pre-training quality and reduce the number of filters to be distilled. The resulting model achieves 10x higher throughput than Transformers and 1.5x higher than Hyena at 1.3B parameters, without any loss in quality after distillation

Propiedades de forma y positividad total en números de Stirling generalizados

En el presente Trabajo de Fin de Máster se realiza un estudio sobre propiedades de forma (log-concavidad) y de positividad total de orden 2 en dos generalizaciones de los números de Stirling de segunda especie. Se presenta una generalización a raíz de la función generatriz que presenta buenas propiedades de forma y positividad total bajo ciertas hipótesis, así como una segunda generalización probabilística que promete presentar buen comportamiento en términos de positividad total, de nuevo bajo algunas presunciones. El objetivo del estudio de dichas generalizaciones es poder deducir propiedades de números bien conocidos en la literatura que son casos particulares de las mismas, como los números de Stirling de segunda especie o los números de Lah, entre otros.El documento se encuentra redactado en inglés. No obstante, se incluye un resumen detallado de la línea argumental del trabajo.<br /