Pointwise Convergence in Probability of General Smoothing Splines

Establishing the convergence of splines can be cast as a variational problem which is amenable to a $\Gamma$-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, $n$, as $\lambda_n=n^{-p}$. Using standard theorems from the $\Gamma$-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for $p\leq \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$

Deep Limits of Residual Neural Networks

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The contribution of this paper is to show that, for the residual neural network model, the deep layer limit coincides with a parameter estimation problem for a nonlinear ordinary differential equation. In particular, whilst it is known that the residual neural network model is a discretisation of an ordinary differential equation, we show convergence in a variational sense. This implies that optimal parameters converge in the deep layer limit. This is a stronger statement than saying for a fixed parameter the residual neural network model converges (the latter does not in general imply the former). Our variational analysis provides a discrete-to-continuum $\Gamma$-convergence result for the objective function of the residual neural network training step to a variational problem constrained by a system of ordinary differential equations; this rigorously connects the discrete setting to a continuum problem

Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using kk-Means Type Optimization

We address the task of estimating multiple trajectories from unlabeled data. This problem arises in many settings, one could think of the construction of maps of transport networks from passive observation of travellers, or the reconstruction of the behaviour of uncooperative vehicles from external observations, for example. There are two coupled problems. The first is a data association problem: how to map data points onto individual trajectories. The second is, given a solution to the data association problem, to estimate those trajectories. We construct estimators as a solution to a regularized variational problem (to which approximate solutions can be obtained via the simple, efficient and widespread $k$-means method) and show that, as the number of data points, $n$, increases, these estimators exhibit stable behaviour. More precisely, we show that they converge in an appropriate Sobolev space in probability and with rate $n^{-1/2}$

Convergence of the kk-Means Minimization Problem using Γ\Gamma-Convergence

The $k$-means method is an iterative clustering algorithm which associates each observation with one of $k$ clusters. It traditionally employs cluster centers in the same space as the observed data. By relaxing this requirement, it is possible to apply the $k$-means method to infinite dimensional problems, for example multiple target tracking and smoothing problems in the presence of unknown data association. Via a $\Gamma$-convergence argument, the associated optimization problem is shown to converge in the sense that both the $k$-means minimum and minimizers converge in the large data limit to quantities which depend upon the observed data only through its distribution. The theory is supplemented with two examples to demonstrate the range of problems now accessible by the $k$-means method. The first example combines a non-parametric smoothing problem with unknown data association. The second addresses tracking using sparse data from a network of passive sensors

Consistency of Fractional Graph-Laplacian Regularization in Semi-Supervised Learning with Finite Labels

Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labelled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimising a graph-Dirichlet energy, equivalently a discrete Sobolev $\mathrm{H}^1$ semi-norm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimisers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimisers are not continuous. One solution is to consider higher-order regularisation, which is the analogue of minimising Sobolev $\mathrm{H}^s$ semi-norms. In this paper we consider the asymptotics of minimising a graph variant of the Sobolev $\mathrm{H}^s$ semi-norm with pointwise constraints. We show that, as expected, one needs $s>d/2$ where $d$ is the dimension of the data manifold. We also show that there must be a upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behaviour of the minimiser even when $s>d/2$.Comment: 37 pages, 4 figure

Large data limit for a phase transition model with the p-Laplacian on point clouds

The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $\epsilon=\epsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\to \infty$, in the regime where $\epsilon_n\to 0$. The mathematical tool used to address this question is $\Gamma$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter