Seminars

In this talk, we offer algorithms to compute relevant sets of generators of equivariants modules, together with generators for the ring of invariants. We show how ideal interpolation can be applied to compute the generating invariants and equivariants of a reflection group. Given a set of primary invariants for any representation of a finite group, we apply interpolation algorithms to compute both a set of secondary invariants; and free bases of all fundamental equivariant modules. We propose a new algorithm to compute a set of generating invariants simultaneously to the generating equivariants.

By definition, a rigid graph in $latex \mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to $\mathbb{C}^d$. A major open problem has been to obtain tight upper bounds on the number of embeddings in $\mathbb{C}^d$, for a given number $|V|$ of vertices, which obviously also bound their number in $\mathbb{R}^d$. Moreover, in most known cases, the actual numbers of embeddings in $\mathbb{R}^d$ and $\mathbb{C}^d$ coincide. For decades, only the trivial bound of $O(2^{d\cdot |V|})$ was known on the number of embeddings. Recently, matrix permanent bounds have led to a small improvement for $d\geq 5$.

This work improves upon the existing upper bounds for the number of embeddings in $\mathbb{C}^d$ and $S^d$, by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively deletes vertices or vertex paths. For the most important cases of $d=2$ and $d=3$, the new bounds are $O(3.7764^{|V|})$ and $O(6.8399^{|V|})$, respectively. In general, the recent asymptotic bound mentioned above is improved by a factor of $1/ \sqrt{2}$. Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic bounds.

In this talk, we introduce a symbolic-numerical algorithm to solve (nearly) degenerate sparse polynomial systems. For that, we consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety X. By working over the Cox ring of X, our algorithm can handle input systems whose solutions have multiplicities. We investigate the regularity of these systems to provide complexity bounds for our approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. The talk is based on joint work with Simon Telen.

We experiment with an unsupervised learning-based method for the classification of 3-dimensional point clouds. The key ingredient in our approach is DeepCluster, a clustering framework that jointly learns the parameters of a neural network and the cluster assignments of the resulting features. We adapt DeepCluster to learn useful feature maps on point clouds, by employing neural networks operating on raw point cloud data without passing to an intermediate regular representation. We pre-train our model across all categories of the ShapeNet dataset. Then, we report the classification accuracy on the ModelNet40 and ModelNet10 datasets, by extracting the features and training a linear SVM classifier on top of these.

With the coming of e-shops, a new trend started spreading among customers:
shopping by home, with no more relying on physical stores. If, on a side,
this can appear useful, on the other side it hides a negative aspect: the re-
turns. In fact, especially for the clothing industry's commodity, it happens
that the size of the purchased articles is wrong or the products do not fit
as desired; this is due to the fact that customers buy them without trying
them on. Lately, these e-shops have been established even by medical health
stores, which sell, for example, masks for sleep apnea equipment. The purpose of my work, then, is to find a criterion to be used during an online purchase to assign a specific kind of apnea mask (i.e. nasal mask, total face mask, oral mask) in such a way it fits the best the buyer's face starting from a 3D mesh of the client's face.

The purpose of this study is to provide means to physicians for automated and fast recognition of airways diseases. In this work, we mainly focus on measures that can be easily recorded using a spirometer. The signals used in this framework are simulated using the linear bi-compartment model of the lungs. This allows us to simulate ventilation under the hypothesis of ventilation at rest (tidal breathing). By changing the resistive and elastic parameters, data samples are realized simulating healthy, fibrosis and asthma breathing. On this synthetic data, different machine learning models are tested and their performance is assessed. All but the Naive bias classifier show accuracy of at least 99%. This represents a proof of concept that Machine Learning can accurately differentiate diseases based on manufactured spirometry data. This paves the way for further developments on the topic, notably testing the model on real data.

A generalized additive decomposition (GAD) of a d-homogeneous polynomial F is a decomposition of this polynomial as a sum of powers of linear forms multiplied by homogeneous polynomials of degree not greater than d. From every GAD of F, there is a natural way of defining a scheme that is apolar to F.
Among all the schemes constructed this way, there is also a cactus scheme to F that may be symbolically detected via an algorithm we have recently proposed.
In this talk, we discuss the d-regularity of such schemes and how this may be employed to better the performance of the aforementioned algorithm.
This is joint work with A. Bernardi and A. Oneto.