Related papers: When is there a Representer Theorem? Reflexive Ban…
We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary…
In this paper, we give a multiplication operator representation of bounded self-adjoint operators T on a Hilbert space H such that -- is a frame for H, for some -- . We state a necessary condition in order for a frame -- to have a…
It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case…
We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$ for…
Describing the solutions of inverse problems arising in signal or image processing is an important issue both for theoretical and numerical purposes. We propose a principle which describes the solutions to convex variational problems…
A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus…
This paper studies existence and uniqueness of solutions to generalized Volterra integral equations. Since our proof for existence and uniqueness does not make use of Banach fixed point theorem unlike the previous papers focused on this…
While conditional diffusion models have achieved remarkable success in various applications, they require abundant data to train from scratch, which is often infeasible in practice. To address this issue, transfer learning has emerged as an…
Recent advances in machine learning have led to increased interest in reproducing kernel Banach spaces (RKBS) as a more general framework that extends beyond reproducing kernel Hilbert spaces (RKHS). These works have resulted in the…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
Motivated by recent work in Dynamical Sampling, we prove a necessary and sufficient condition for a frame in a separable and infinite-dimensional Hilbert space to admit the form $\{T^{n} \varphi \}_{n \geq 0}$ with $T \in B(H)$. Also, a…
We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however,…
A common assumption about neural networks is that they can learn an appropriate internal representations on their own, see e.g. end-to-end learning. In this work we challenge this assumption. We consider two simple tasks and show that the…
We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A…
The interpretation of spatial references is highly contextual, requiring joint inference over both language and the environment. We consider the task of spatial reasoning in a simulated environment, where an agent can act and receive…
Representation learning is a central challenge across a range of machine learning areas. In reinforcement learning, effective and functional representations have the potential to tremendously accelerate learning progress and solve more…
Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a…
Learning representations of data is an important problem in statistics and machine learning. While the origin of learning representations can be traced back to factor analysis and multidimensional scaling in statistics, it has become a…
Learning representations for pixel-based control has garnered significant attention recently in reinforcement learning. A wide range of methods have been proposed to enable efficient learning, leading to sample complexities similar to those…
The minimum-norm interpolator (MNI) framework has recently attracted considerable attention as a tool for understanding generalization in overparameterized models, such as neural networks. In this work, we study the MNI under a $2$-uniform…