On the Equivalence of f-Divergence Balls and Density Bands in Robust Detection

The paper deals with minimax optimal statistical tests for two composite hypotheses, where each hypothesis is defined by a non-parametric uncertainty set of feasible distributions. It is shown that for every pair of uncertainty sets of the f-divergence ball type, a pair of uncertainty sets of the density band type can be constructed, which is equivalent in the sense that it admits the same pair of least favorable distributions. This result implies that robust tests under $f$-divergence ball uncertainty, which are typically only minimax optimal for the single sample case, are also fixed sample size minimax optimal with respect to the equivalent density band uncertainty sets.Comment: 5 pages, 1 figure, accepted for publication in the Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 201

Frontiers in Nonparametric Statistics

The goal of this workshop was to discuss recent developments of nonparametric statistical inference. A particular focus was on high dimensional statistics, semiparametrics, adaptation, nonparametric bayesian statistics, shape constraint estimation and statistical inverse problems. The close interaction of these issues with optimization, machine learning and inverse problems has been addressed as well

Doctor of Philosophy

dissertationWith modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming in- creasingly accessible. As a result, the size and complexity of data representing such phenom- ena are also increasing, making the role of data analysis to propel science even more integral. This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields--an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single ""correct"" reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of ""correctness"" of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (time- independent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Entanglement and Bell correlations in strongly correlated many-body quantum systems

During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources. With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems. Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices. While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos últimas décadas, gracias al enriquecimiento mutuo entre las investigaciones en información cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy útiles en la caracterización de los sistemas cuánticos de muchos cuerpos (SCMC), basados en características puramente cuánticas sin ningún análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en última instancia discriminan los regímenes clásicos de los regímenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plétora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retículos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topología y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numérico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el célebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retículos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posición central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el éxito de los TN depende principalmente de su cumplimiento, por construcción, de una “ley del área” (“area-law”) de la entropía de entrelazamiento. Esta es una característica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropía de entrelazamiento subextensiva, que crece sólo con la superficie de la bipartición. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducción exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados útiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivación de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sólo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capítulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transición de fase de un modelo de Ising ferromagnético con interacciones de largo alcance, haciendo una comparación con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad

Adaptive Statistical Inference

This workshop in mathematical statistics highlights recent advances in adaptive methods for statistical estimation, testing and confidence sets. Related open mathematical problems are discussed with potential impact on the development of computationally efficient algorithms of data processing under prior uncertainty. Parcticular emphasis is on high dimensional models, inverse problems and discrtete structures

Entanglement and Bell correlations in strongly correlated many-body quantum systems

During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources. With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems. Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices. While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos últimas décadas, gracias al enriquecimiento mutuo entre las investigaciones en información cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy útiles en la caracterización de los sistemas cuánticos de muchos cuerpos (SCMC), basados en características puramente cuánticas sin ningún análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en última instancia discriminan los regímenes clásicos de los regímenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plétora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retículos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topología y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numérico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el célebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retículos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posición central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el éxito de los TN depende principalmente de su cumplimiento, por construcción, de una “ley del área” (“area-law”) de la entropía de entrelazamiento. Esta es una característica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropía de entrelazamiento subextensiva, que crece sólo con la superficie de la bipartición. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducción exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados útiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivación de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sólo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capítulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transición de fase de un modelo de Ising ferromagnético con interacciones de largo alcance, haciendo una comparación con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad.Postprint (published version