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We develop a general framework for working with structured lifting problems, establishing closure and uniqueness properties of their solutions. In a subsequent paper, we apply these results to axiomatize computation rules of cubical type…
The human ability to learn, generalize, and control complex manipulation tasks through multi-modality feedback suggests a unique capability, which we refer to as dexterity intelligence. Understanding and assessing this intelligence is a…
Crisp and lattice-valued ambiguous representations of one continuous semilattice in another one are introduced and operation of taking pseudo-inverse of the above relations is defined. It is shown that continuous semilattices and their…
The linear representation hypothesis states that language models (LMs) encode concepts as directions in their latent space, forming organized, multidimensional manifolds. Prior work has largely focused on identifying specific geometries for…
Recent work utilizes Large Language Models (LLMs) for topic modeling, generating comprehensible topic labels for given documents. However, their performance has mainly been evaluated qualitatively, and there remains room for quantitative…
Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern…
Language models (LMs) are increasingly used as simulacra for people, yet their ability to match the distribution of views of a specific demographic group and be \textit{distributionally aligned} remains uncertain. This notion of…
Trained ML models are commonly embedded in optimization problems. In many cases, this leads to large-scale NLPs that are difficult to solve to global optimality. While ML models frequently lead to large problems, they also exhibit…
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…
Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for…
With the continuous advancement of large language models (LLMs), it is essential to create new benchmarks to effectively evaluate their expanding capabilities and identify areas for improvement. This work focuses on multi-image reasoning,…
Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and…
It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional…
Mathematically representing the shape of an object is a key ingredient for solving inverse rendering problems. Explicit representations like meshes are efficient to render in a differentiable fashion but have difficulties handling topology…
In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from…
Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it will prove instrumental in detecting…
Mutual information (MI) is a general measure of statistical dependence with widespread application across the sciences. However, estimating MI between multi-dimensional variables is challenging because the number of samples necessary to…
Linear approximations to the decision boundary of a complex model have become one of the most popular tools for interpreting predictions. In this paper, we study such linear explanations produced either post-hoc by a few recent methods or…
Starting from a linear fractional representation of a linear system affected by constant parametric uncertainties, we demonstrate how to enhance standard robust analysis tests by taking available (noisy) input-output data of the uncertain…
We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian…