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The power of multivariate functions is their ability to model a wide variety of phenomena, but have the disadvantages that they lack an intuitive or interpretable representation, and often require a (very) large number of parameters. We…
Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators…
Aspects of the theory of characteristic modes, based on their variational formulation, are presented and an explicit form of a related functional, involving only currents in a spatial domain, is derived. The new formulation leads to deeper…
Many fMRI analyses examine functional connectivity, or statistical dependencies among remote brain regions. Yet popular methods for studying whole-brain functional connectivity often yield results that are difficult to interpret. Factor…
We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified.…
Despite the ubiquity of tabular data in high-stakes domains, traditional deep learning architectures often struggle to match the performance of gradient-boosted decision trees while maintaining scientific interpretability. Standard neural…
We present a formal and constructive simulation framework for nondeterministic finite automata (NFAs) using time-shared, depth-unrolled feedforward networks (TS-FFNs), i.e., acyclic unrolled computations with shared parameters that are…
Neurons are the fundamental building blocks of deep neural networks, and their interconnections allow AI to achieve unprecedented results. Motivated by the goal of understanding how neurons encode information, compositional explanations…
Using representations of functional data can be more convenient and beneficial in subsequent statistical models than direct observations. These representations, in a lower-dimensional space, extract and compress information from individual…
Function is defined as the ensemble of tasks that enable the product to complete the designed purpose. Functional tools, such as functional modeling, offer decision guidance in the early phase of product design, where explicit design…
The development of data acquisition systems is facilitating the collection of data that are apt to be modelled as functional data. In some applications, the interest lies in the identification of significant differences in group functional…
We propose a functional view of matrix decomposition problems on graphs such as geometric matrix completion and graph regularized dimensionality reduction. Our unifying framework is based on the key idea that using a reduced basis to…
Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we…
We propose the Decomposer Networks (DecompNet), a semantic autoencoder that factorizes an input into multiple interpretable components. Unlike classical autoencoders that compress an input into a single latent representation, the Decomposer…
As deep neural models in NLP become more complex, and as a consequence opaque, the necessity to interpret them becomes greater. A burgeoning interest has emerged in rationalizing explanations to provide short and coherent justifications for…
Structured recursion schemes such as folds and unfolds have been widely used for structuring both functional programs and program semantics. In this context, it has been customary to implement denotational semantics as folds over an…
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important…
Some scientific publications are under suspicion of fabrication of data. Since humans are bad random number generators, there might be some evidential value in favor of fabrication in the statistical results as presented in such papers. In…
This paper introduces a framework for modeling cyclical and feedback-driven information flow through a generalized family of entropy-modulated transformations called derangetropy functionals. Unlike scalar and static entropy measures such…
A common problem in various applications is the additive decomposition of the output of a function with respect to its input variables. Functions with binary arguments can be axiomatically decomposed by the famous Shapley value. For the…