Producto cuántico entre funciones simétricas

Las funciones simétricas presentan resultados de interés para diversas áreas de la matemática. Una generalización de estos objetos son las funciones multi simétricas, que serán el objeto de estudio del presente trabajo. El espacio de funciones multi simétricas puede considerarse como el espacio de funciones en el cociente (R)n/sn, que hereda una estructura de variedad de Poisson. Debido a este último hecho (R)n/sn, permite una cuantización por deformación, que en nuestro contexto veremos como la posibilidad de definir un producto estrella asociativo y no conmutativo, denotado *. Este producto, trabajado por Diaz y Pariguan, permite una descripción explícita y su cálculo involucra varias propiedades combinatorias de interés. En el trabajo se incluyen secciones sobre funciones simétricas, geometría simplectica y de Poisson y cuantización por deformación. Como producto del trabajo se creó un programa para calcular y dar ejemplos de este producto estrella y se lograron encontrar mejoras para el cálculo del mismo. Se muestran el código desarrollado, ejercicios de simulación de tiempo de ejecución y también conjeturas que nacieron como conclusión del trabajo.The symmetric functions appear as important study objects in different areas of mathematics. An immediate generalization of these objects is the space of multi symmetric functions, which will be a main item in the present work. The space of multi symmetric functions can be considered as the space of functions in the quotient space (R)n/sn, which inherits a Poisson manifold structure. Because of this fact, (R)n/sn, allows a deformation quantization that will be understood through this work as the possibility of defining a star product which is associative but non commutative denoted ?. Diaz and Pariguan introduced the explicit formulation of this product, and its calculation involves some interesting combinatorics. A computer program for developing this star product was created as a result of the present work. It was possible to give examples of the product and improvements for the formula. The program code is shown with some examples made to study the computer time consumed by the execution of the program. We also show some conjectures that were born during the development of the work.Matemático (a)Pregrad

Randomization for adversarial robustness: the Good, the Bad and the Ugly

Deep neural networks are known to be vulnerable to adversarial attacks: A small perturbation that is imperceptible to a human can easily make a well-trained deep neural network misclassify. To defend against adversarial attacks, randomized classifiers have been proposed as a robust alternative to deterministic ones. In this work we show that in the binary classification setting, for any randomized classifier, there is always a deterministic classifier with better adversarial risk. In other words, randomization is not necessary for robustness. In many common randomization schemes, the deterministic classifiers with better risk are explicitly described: For example, we show that ensembles of classifiers are more robust than mixtures of classifiers, and randomized smoothing is more robust than input noise injection. Finally, experiments confirm our theoretical results with the two families of randomized classifiers we analyze.Comment: 8 pages + bibliography and appendix, 3 figures. Submitted to ICML 202

Adversarial attacks for mixtures of classifiers

Mixtures of classifiers (a.k.a. randomized ensembles) have been proposed as a way to improve robustness against adversarial attacks. However, it has been shown that existing attacks are not well suited for this kind of classifiers. In this paper, we discuss the problem of attacking a mixture in a principled way and introduce two desirable properties of attacks based on a geometrical analysis of the problem (effectiveness and maximality). We then show that existing attacks do not meet both of these properties. Finally, we introduce a new attack called lattice climber attack with theoretical guarantees on the binary linear setting, and we demonstrate its performance by conducting experiments on synthetic and real datasets.Comment: 7 pages + 4 pages of appendix. 5 figures in main tex