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Benny Avelin

Publications and source records attributed to Benny Avelin.

32 records · Page 2Linked to original sources
arXiv2021

Uncertainty-Aware Body Composition Analysis with Deep Regression Ensembles on UK Biobank MRI

Along with rich health-related metadata, medical images have been acquired for over 40,000 male and female UK Biobank participants, aged 44-82, since 2014. Phenotypes derived from these images, such as measurements of body composition from MRI, can reveal new links between genetics, cardiovascular disease, and metabolic conditions. In this work, six measurements of body composition and adipose tissues were automatically estimated by image-based, deep regression with ResNet50 neural networks from neck-to-knee body MRI. Despite the potential for high speed and accuracy, these networks produce no output segmentations that could indicate the reliability of individual measurements. The presented experiments therefore examine uncertainty quantification with mean-variance regression and ensembling to estimate individual measurement errors and thereby identify potential outliers, anomalies, and other failure cases automatically. In 10-fold cross-validation on data of about 8,500 subjects, mean-variance regression and ensembling showed complementary benefits, reducing the mean absolute error across all predictions by 12%. Both improved the calibration of uncertainties and their ability to identify high prediction errors. With intra-class correlation coefficients (ICC) above 0.97, all targets except the liver fat content yielded relative measurement errors below 5%. Testing on another 1,000 subjects showed consistent performance, and the method was finally deployed for inference to 30,000 subjects with missing reference values. The results indicate that deep regression ensembles could ultimately provide automated, uncertainty-aware measurements of body composition for more than 120,000 UK Biobank neck-to-knee body MRI that are to be acquired within the coming years.

eess.IV↗
arXiv2021

Deep limits and cut-off phenomena for neural networks

We consider dynamical and geometrical aspects of deep learning. For many standard choices of layer maps we display semi-invariant metrics which quantify differences between data or decision functions. This allows us, when considering random layer maps and using non-commutative ergodic theorems, to deduce that certain limits exist when letting the number of layers tend to infinity. We also examine the random initialization of standard networks where we observe a surprising cut-off phenomenon in terms of the number of layers, the depth of the network. This could be a relevant parameter when choosing an appropriate number of layers for a given learning task, or for selecting a good initialization procedure. More generally, we hope that the notions and results in this paper can provide a framework, in particular a geometric one, for a part of the theoretical understanding of deep neural networks.

cs.LG↗
arXiv2021

Remarks on existence and uniqueness of the solution for stochastic partial differential equations

In this article we consider existence and uniqueness of the solutions to a large class of stochastic partial differential of form $\partial_t u = L_x u + b(t,u)+σ(t,u)\dot{W}$, driven by a Gaussian noise $\dot{W}$, white in time and spatial correlations given by a generic covariance $γ$. We provide natural conditions under which classical Picard iteration procedure provides a unique solution. We illustrate the applicability of our general result by providing several interesting particular choices for the operator $L_x$ under which our existence and uniqueness result hold. In particular, we show that Dalang condition given in Dalang (1999) is sufficient in the case of many parabolic and hypoelliptic operators $L_x$.

math.PR↗
arXiv2021

Approximation of BV functions by neural networks: A regularity theory approach

In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincaré inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.

math.AP↗
arXiv2020

Boundary behavior of solutions to the parabolic p-Laplace equation II

This paper is the second installment in a series of papers concerning the boundary behavior of solutions to the $p$-parabolic equations. In this paper we are interested in the short time behavior of the solutions, which is in contrast with much of the literature, where all results require a waiting time. We prove a dichotomy about the decay-rate of non-negative solutions vanishing on the lateral boundary in a cylindrical $C^{1,1}$ domain. Furthermore we connect this dichotomy to the support of the boundary type Riesz measure related to the $p$-parabolic equation in NTA-domains, which has consequences for the continuation of solutions.

math.AP↗
arXiv2020

Neural ODEs as the Deep Limit of ResNets with constant weights

In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.

stat.ML↗
arXiv2015

Boundary behavior of solutions to the parabolic p-Laplace equation

We establish boundary estimates for non-negative solutions to the p-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

math.AP↗
arXiv2015

Characterizations of interior polar sets for the degenerate p-parabolic equation

This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.

math.AP↗
arXiv2015

A comparison principle for the porous medium equation and its consequences

We prove a comparison principle for the porous medium equation in more general open sets in $\mathbb{R}^{n+1}$ than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

math.AP↗
arXiv2014

Boundary Estimates for Certain Degenerate and Singular Parabolic Equations

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.

math.AP↗
arXiv2013

Lower semicontinuity of weak supersolutions to the porous medium equation

Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that nonnegative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.

math.AP↗
arXiv2013

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $μ$. Our estimates are derived assuming that (i) the control distance $d$ generated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the $μ$-measure of $d-$metric balls and (iii) the validity of a Poincaré inequality involving $\X$ and $μ$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

math.AP↗
arXiv2012

Approximation and Bounded Plurisubharmonic Exhaustion Functions Beyond Lipschitz Domains

Using techniques from the analysis of PDEs to study the boundary behaviour of functions on domains with low boundary regularity, we extend results by Fornaæss-Wiegerinck (1989) on plurisubharmonic approximation and by Demailly (1987) on the existence on bounded plurisubharmonic exhaustion functions to domains beyond Lipschitz boundary regularity.

math.CV↗