arXiv subjects

Lukáš Malý

Publications and source records attributed to Lukáš Malý.

11 recordsLinked to original sources

arXiv2025

Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces

We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices $X$ as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case $X=L^p$ with $1\le p<\infty$, where all Sobolev-type functions in $N^p$ are known to be quasicontinuous, provided that the underlying metric space $\mathcal{P}$ is locally complete. In most of our examples, $\mathcal{P}$ is a compact subset of $\mathbf{R}^2$ and $X=L^\infty$. Four particular examples are the damped topologist's sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali--Carathéodory property.

math.FA↗

arXiv2023

A simple proof of reflexivity and separability of $N^{1,p}$ Sobolev spaces

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$-Poincaré inequality, then the $N^{1,p}(X)$ Sobolev space is reflexive and separable whenever $p\in (1,\infty)$. We also prove separability of the space when $p=1$. Our proof is based on a straightforward construction of an equivalent norm on $N^{1,p}(X)$, $p\in [1,\infty)$, that is uniformly convex when $p\in (1,\infty)$. Finally, we explicitly construct a functional that is pointwise comparable to the minimal $p$-weak upper gradient, when $p\in (1,\infty)$.

math.FA↗

arXiv2022

Standalone Neural ODEs with Sensitivity Analysis

This paper presents the Standalone Neural ODE (sNODE), a continuous-depth neural ODE model capable of describing a full deep neural network. This uses a novel nonlinear conjugate gradient (NCG) descent optimization scheme for training, where the Sobolev gradient can be incorporated to improve smoothness of model weights. We also present a general formulation of the neural sensitivity problem and show how it is used in the NCG training. The sensitivity analysis provides a reliable measure of uncertainty propagation throughout a network, and can be used to study model robustness and to generate adversarial attacks. Our evaluations demonstrate that our novel formulations lead to increased robustness and performance as compared to ResNet models, and that it opens up for new opportunities for designing and developing machine learning with improved explainability.

cs.LG↗

arXiv2022

Learning via nonlinear conjugate gradients and depth-varying neural ODEs

The inverse problem of supervised reconstruction of depth-variable (time-dependent) parameters in a neural ordinary differential equation (NODE) is considered, that means finding the weights of a residual network with time continuous layers. The NODE is treated as an isolated entity describing the full network as opposed to earlier research, which embedded it between pre- and post-appended layers trained by conventional methods. The proposed parameter reconstruction is done for a general first order differential equation by minimizing a cost functional covering a variety of loss functions and penalty terms. A nonlinear conjugate gradient method (NCG) is derived for the minimization. Mathematical properties are stated for the differential equation and the cost functional. The adjoint problem needed is derived together with a sensitivity problem. The sensitivity problem can estimate changes in the network output under perturbation of the trained parameters. To preserve smoothness during the iterations the Sobolev gradient is calculated and incorporated. As a proof-of-concept, numerical results are included for a NODE and two synthetic datasets, and compared with standard gradient approaches (not based on NODEs). The results show that the proposed method works well for deep learning with infinite numbers of layers, and has built-in stability and smoothness.

cs.LG↗

arXiv2017

Trace and extension theorems for Sobolev-type functions in metric spaces

Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-$θ$ bound are considered. Based on a Poincaré inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds $θ$. The trace $T$ is shown to be a compact operator mapping a Sobolev-type space on a domain into a Besov space on the boundary. Sufficient conditions for $T$ to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to $θ$, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in $L^θ$ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.

math.MG↗

arXiv2016

Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity

We employ a variational approach to study the Neumann boundary value problem for the $p$-Laplacian on bounded smooth-enough domains in the metric setting, and show that solutions exist and are bounded. The boundary data considered are Borel measurable bounded functions. We also study boundary continuity properties of the solutions. One of the key tools utilized is the trace theorem for Newton-Sobolev functions, and another is an analog of the De Giorgi inequality adapted to the Neumann problem.

math.MG↗

arXiv2016

Trace and extension theorems for functions of bounded variation

In this paper we show that every $L^1$-integrable function on $\partialΩ$ can be obtained as the trace of a function of bounded variation in $Ω$ whenever $Ω$ is a domain with regular boundary $\partialΩ$ in a doubling metric measure space. In particular, the trace class of $BV(Ω)$ is $L^1(\partialΩ)$ provided that $Ω$ supports a 1-Poincaré inequality. We also construct a bounded linear extension from a Besov class of functions on $\partialΩ$ to $BV(Ω)$.

math.MG↗

arXiv2014

Newtonian spaces based on quasi-Banach function lattices

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

math.FA↗

arXiv2014

Regularization of Newtonian functions via weak boundedness of maximal operators

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a $p$-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on rearrangement-invariant spaces are established and applied.

math.FA↗

arXiv2014

Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (Hölder) continuous. Particular focus is put on the borderline case when the degree of integrability equals the "dimension of the measure". If Lipschitz functions are dense in a Newtonian space on a proper metric space, then locally Lipschitz functions are proven dense in the corresponding Newtonian space on open subsets, where no hypotheses (besides being open) are put on these sets.

math.FA↗

arXiv2012

Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices

Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

math.FA↗