We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We prove that an optimal solution to the inverse problem is a closed curve with spline components. We then show how to efficiently solve the task using B-splines as basis functions. We extend our problem formulation to curves made of two distinct components with complementary smoothness properties and solve it using hybrid splines. We illustrate the performance of our model on contours of different smoothness. Our experimental results show that we can faithfully reconstruct any general contour using few parameters, even in the presence of imprecisions in the measurements.

In this paper, we precisely quantify the wavelet compressibility of compound Poisson processes. To that end, we expand the given random process over the Haar wavelet basis and we analyse its asymptotic approximation properties. By only considering the nonzero wavelet coefficients up to a given scale, what we call the greedy approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewise-constant nature of compound Poisson processes. More precisely, we provide lower and upper bounds for the mean squared error of greedy approximation of compound Poisson processes. We are then able to deduce that the greedy approximation error has a sub-exponential and super-polynomial asymptotic behavior. Finally, we provide numerical experiments to highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.

The motivation for this work is to improve the performance of deep neural networks through the optimization of the individual activation functions. Since the latter results in an infinite- dimensional optimization problem, we resolve the ambiguity by searching for the sparsest and most regular solution in the sense of Lipschitz. To that end, we first introduce a bound that relates the properties of the pointwise nonlinearities to the global Lipschitz constant of the network. By using the proposed bound as regularizer, we then derive a representer theorem that shows that the optimum configuration is achievable by a deep spline network. It is a variant of a conventional deep ReLU network where each activation function is a piecewise-linear spline with adaptive knots. The practical interest is that the underlying spline activations can be expressed as linear combinations of ReLU units and optimized using ℓ1-minimization techniques.