Anomaly of (2+1)(2+1)-Dimensional Symmetry-Enriched Topological Order from (3+1)(3+1)-Dimensional Topological Quantum Field Theory

Symmetry acting on a (2+1)$D$ topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)$D$ on-site symmetry. In this paper, we develop a (3+1)$D$ topological quantum field theory to calculate the anomaly indicators of a (2+1)$D$ topological order with a general symmetry group $G$, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)$D$ topological quantum field theory on a specific manifold equipped with some $G$-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including $\mathbb{Z}_2\times\mathbb{Z}_2$, $\mathbb{Z}_2^T\times\mathbb{Z}_2^T$, $SO(N)$, $O(N)^T$, $SO(N)\times \mathbb{Z}_2^T$, etc, where $\mathbb{Z}_2$ and $\mathbb{Z}_2^T$ denote a unitary and anti-unitary order-2 group, respectively, and $O(N)^T$ denotes a symmetry group $O(N)$ such that elements in $O(N)$ with determinant $-1$ are anti-unitary. In particular, we demonstrate that some anomaly of $O(N)^T$ and $SO(N)\times \mathbb{Z}_2^T$ exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for $SO(N)$ symmetric topological orders, we derive their $SO(N)$ Hall conductance.Comment: Recipe of calculating the partition function involving continuous symmetries added, together with extra example

How to address monotonicity for model risk management?

In this paper, we study the problem of establishing the accountability and fairness of transparent machine learning models through monotonicity. Although there have been numerous studies on individual monotonicity, pairwise monotonicity is often overlooked in the existing literature. This paper studies transparent neural networks in the presence of three types of monotonicity: individual monotonicity, weak pairwise monotonicity, and strong pairwise monotonicity. As a means of achieving monotonicity while maintaining transparency, we propose the monotonic groves of neural additive models. As a result of empirical examples, we demonstrate that monotonicity is often violated in practice and that monotonic groves of neural additive models are transparent, accountable, and fair

Aspects of Anomaly in Condensed Matter Physics

Anomalies have proven to be an important tool for unraveling the enigmatic properties of strongly-coupled condensed matter systems. Especially, it provides powerful constraints on the emergibility problem, i.e., whether a quantum phase or phase transition can emerge in a many-body system. In this thesis, the focus is mainly on $(2+1)$-dimensional bosonic systems, and we explain how to identify the anomaly of various interesting systems in this category. We then apply these results to study the emergibility of a class of quantum critical states known as Stiefel Liquid, and discuss several interesting realizations uncovered using results from anomaly. First of all, for a $(2+1)$-d lattice spin system, we derive the topological partition functions that characterize the Lieb-Schultz-Mattis constraints with $G_s\times G_{int}$ symmetry, where $G_s$ is an arbitrary space group in two spatial dimensions, and $G_{int}$ is any internal symmetry whose projective representations are classified by $(\mathbb{Z}_2)^k$ with $k$ an integer. This LSM anomaly will be matched with the anomaly of IR states. We then calculate the anomaly of several IR states. One class of IR states we discuss is the recently-proposed Stiefel Liquid, with the well-known Deconfined Quantum Critical Point (DQCP) and U(1) Dirac Spin Liquid (DSL) unified as two special examples. We introduce the description of Stiefel Liquid using non-linear Sigma Model and explain how to get the anomaly from such description. Another class of IR states we consider is $(2+1)$-d symmetry-enriched topological order. And we explain the framework of getting the anomaly of topological order with a general symmetry action by identifying a $(3+1)$-d topological quantum field theory whose boundary hosts the original $(2+1)$-d symmetry-enriched topological order. Finally, we apply these results and the framework of anomaly-matching to understand the emergibility of Stiefel Liquid, including DQCP, DSL, and the so-called non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with $SO(3) \times \mathbb{Z}_2^T$ internal symmetry and either $p6m$ or $p4m$ lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spinquadrupolar liquids of which the most relevant spinful fluctuations carry spin-2. In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction

Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality

Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the emergibility problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with $G_s\times G_{int}$ symmetry, where $G_s$ is an arbitrary space group in one or two spatial dimensions, and $G_{int}$ is any internal symmetry whose projective representations are classified by $\mathbb{Z}_2^k$ with $k$ an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the well-known deconfined quantum critical point (DQCP), $U(1)$ Dirac spin liquid (DSL), and the recently proposed non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with $SO(3)\times \mathbb{Z}_2^T$ internal symmetry and either $p6m$ or $p4m$ lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spin-quadrupolar liquids of which the most relevant spinful fluctuations carry spin-$2$. In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO$_2$.Comment: 23 pages of main text + appendices + ancillary file

Aberrant Dynamic Functional Connectivity of Posterior Cingulate Cortex Subregions in Major Depressive Disorder With Suicidal Ideation

Accumulating evidence indicates the presence of structural and functional abnormalities of the posterior cingulate cortex (PCC) in patients with major depressive disorder (MDD) with suicidal ideation (SI). Nevertheless, the subregional-level dynamic functional connectivity (dFC) of the PCC has not been investigated in MDD with SI. We therefore sought to investigate the presence of aberrant dFC variability in PCC subregions in MDD patients with SI. We analyzed resting-state functional magnetic resonance imaging (fMRI) data from 31 unmedicated MDD patients with SI (SI group), 56 unmedicated MDD patients without SI (NSI group), and 48 matched healthy control (HC) subjects. The sliding-window method was applied to characterize the whole-brain dFC of each PCC subregion [the ventral PCC (vPCC) and dorsal PCC (dPCC)]. In addition, we evaluated associations between clinical variables and the aberrant dFC variability of those brain regions showing significant between-group differences. Compared with HCS, the SI and the NSI groups exhibited higher dFC variability between the left dPCC and left fusiform gyrus and between the right vPCC and left inferior frontal gyrus (IFG). The SI group showed higher dFC variability between the left vPCC and left IFG than the NSI group. Furthermore, the dFC variability between the left vPCC and left IFG was positively correlated with Scale for Suicidal Ideation (SSI) score in patients with MDD (i.e., the SI and NSI groups). Our results indicate that aberrant dFC variability between the vPCC and IFG might provide a neural-network explanation for SI and may provide a potential target for future therapeutic interventions in MDD patients with SI

Tembusu Virus in Ducks, China

In China in 2010, a disease outbreak in egg-laying ducks was associated with a flavivirus. The virus was isolated and partially sequenced. The isolate exhibited 87%–91% identity with strains of Tembusu virus, a mosquito-borne flavivirus of the Ntaya virus group. These findings demonstrate emergence of Tembusu virus in ducks

Bandit Methods and Selective Prediction in Deep Learning

his thesis consists of three parts involving two areas in the field of machine learning: deep learning and reinforcement learning. Stochastic Lipschitz bandit algorithms are methods that govern exploration-exploitation trade-offs and have been used for a variety of important task domains, including zeroth-order optimization. In the first part, we present a framework for Lipschitz bandit methods that adaptively learns partitions of context- and arm-space. Due to this flexibility, the algorithm can efficiently optimize rewards and minimize regret, by focusing on the most relevant portionsof the space. Tuning hyperparameters is a crucial problem in deep learning that is required in every single model. Although the deep neural network with high complexity can fit the large-scale data very well, it uses plenty of hyperparameters. Our algorithms can achieve stateof-the-art performance in challenging real-world tasks such as neural network hyperparameter tuning. Model-free Reinforcement Learning algorithms, such as Q-learning, require less space and are more expressive to use compared to model-based approaches. Upper Confidence Bound (UCB) exploration policy improves the exploration bonuses in the Q-learning framework. In the second part of this work, we study the regret bound of the Q-learning algorithm with UCB exploration in the scenario of compact state-action metric space. We develop a Q-learningalgorithm with UCB exploration policy that adaptively discretizes the continuous state-action space and iteratively updates Q-values. In addition, the algorithm can efficiently optimize rewards and minimize the cumulative regret. We provide a rigorous analysis of bounding the regret with concentration of measure analysis and combinatorial approaches. This is the first result of this kind to the best of our knowledge. Data gathered from real-world applications often suffer from corruption. The low-quality data will hinder the performance of the learning system in terms of the classification accuracy, model building time, and interpretability of the classifier. Selective prediction, also known as prediction with a reject option, reduces the error rate by abstaining from prediction under uncertainty while keeping coverage as high as possible. In the third part of this work, we propose sophisticated threshold learning algorithms integrated with selective prediction thatcan estimate the intrinsic rejection rate of the dataset. Correspondingly, we provide a rigorous framework to generalize the estimation of data corruption rate. To leverage the advantage of multiple learning methods, we extend learning algorithms to a hierarchical two-stage system. Our methods have advantages of being noise-robust, intelligible, and flexible with any networkarchitecture. The empirical results show that our algorithms can accomplish state-of-the-art performance in real-world datasets in many classification and regression problems.</div

Analysis of Performance Measure in Q Learning with UCB Exploration

Compared to model-based Reinforcement Learning (RL) approaches, model-free RL algorithms, such as Q-learning, require less space and are more expressive, since specifying value functions or policies is more flexible than specifying the model for the environment. This makes model-free algorithms more prevalent in modern deep RL. However, model-based methods can more efficiently extract the information from available data. The Upper Confidence Bound (UCB) bandit can improve the exploration bonuses, and hence increase the data efficiency in the Q-learning framework. The cumulative regret of the Q-learning algorithm with an UCB exploration policy in the episodic Markov Decision Process has recently been explored in the underlying environment of finite state-action space. In this paper, we study the regret bound of the Q-learning algorithm with UCB exploration in the scenario of compact state-action metric space. We present an algorithm that adaptively discretizes the continuous state-action space and iteratively updates Q-values. The algorithm is able to efficiently optimize rewards and minimize cumulative regret