# Seminars & Events for PACM IDeAS

##### Towards de-mystification of deep learning: function space analysis of the representation layers

**PLEASE NOTE DIFFERENT DAY (TUESDAY).** We propose a function space approach to Representation Learning [1] and the analysis of the representation layers in deep learning architectures. We show how to compute a `weak-type' Besov smoothness index that quantifies the geometry of the clustering in the feature space. This approach was already applied successfully to improve the performance of machine learning algorithms such as the Random Forest [2] and tree-based Gradient Boosting [3]. Our experiments demonstrate that in well-known and well-performing trained networks, the Besov smoothness of the training set, measured in the corresponding hidden layer feature map representation, increases from layer to layer which relates to the `unfolding' of the clustering in the feature space.

##### Stability of some super-resolution problems

The problem of computational super-resolution asks to recover an object from its noisy and limited spectrum. In this talk, we consider two inverse problems of this flavor, mainly from the point of view of stability estimates. In the first problem, we assume that the object's spectrum is a finite sum of exponentials modulated by polynomials (extending the well-researched case where the polynomials are constants). We derive upper bounds on the problem condition number and show that the attainable resolution exhibits Hölder-type continuity with respect to the noise level [1,3]. As an application we consider the approximation of a piecewise-smooth function from its Fourier coefficients.

##### TBA - Veit Elser

##### Iron Age Hebrew Epigraphy in the Silicon Age - An Algorithmic Approach To Study Paleo-Hebrew Inscriptions

Handwriting comparison and identification, e.g. in the setting of forensics, has been widely addressed over the years. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. For historical documents, such problems are worsened, due to the inscriptions’ preservation conditions. In the following lecture, we will present an attempt at addressing such a problem in the setting of First Temple Period inscriptions, stemming from the isolated military outpost of Arad (ca. 600 BCE).

##### TBA - Nicolas Garcia Trillos

##### Provably good convex methods for mapping problems

Computing mappings or correspondences between surfaces is an important tool for many applications in computer graphics, computer vision, medical imaging, morphology and related fields. Mappings of least angle distortion (conformal) and distance distortion (isometric) are of particular interest. The problem of finding conformal/isometric mappings between surfaces is typically formulated as a difficult non-convex optimization problem. Convex methods relax the non-convex optimization problem to a convex problem which can then be solved globally. The main issue then is whether the global solution of the convex problem is a good approximation for the original global solution. In this talk we will discuss two families of convex relaxations.

##### IDeAS Seminar: Computational Algebraic Geometry and Applications to Computer Vision

Many models in science and engineering are described by polynomials. Computational algebraic geometry gives tools to analyze and exploit algebraic structure. In this talk, we offer a user-friendly introduction to some of these notions, including dimension (formalizing degrees of freedom), degree (formalizing the number of solutions to a polynomial system) and 0-1 laws in algebraic geometry (solution sets to polynomial systems exhibit similar behavior for all but a measure 0 subset of problem instances). We will also mention algorithms, based on Gröbner bases (symbolic techniques) and homotopy continuation (numerical techniques).

Applied examples are drawn from the structure-from-motion problem in computer vision, where the task of building a 3D model from multiple 2D images leads to nontrivial polynomial systems.