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We study the problem of robust multivariate polynomial regression: let $p\colon\mathbb{R}^n\to\mathbb{R}$ be an unknown $n$-variate polynomial of degree at most $d$ in each variable. We are given as input a set of random samples $(\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R}$ that are noisy versions of $(\mathbf{x}_i,p(\mathbf{x}_i))$. More precisely, each $\mathbf{x}_i$ is sampled independently from some distribution $\chi$ on $[-1,1]^n$, and for each $i$ independently, $y_i$ is arbitrary (i.e., an outlier) with probability at most $\rho < 1/2$, and otherwise satisfies $|y_i-p(\mathbf{x}_i)|\leq\sigma$. The goal is to output a polynomial $\hat{p}$, of degree at most $d$ in each variable, within an $\ell_\infty$-distance of at most $O(\sigma)$ from $p$. Kane, Karmalkar, and Price [FOCS'17] solved this problem for $n=1$. We generalize their results to the $n$-variate setting, showing an algorithm that achieves a sample complexity of $O_n(d^n\log d)$, where the hidden constant depends on $n$, if $\chi$ is the $n$-dimensional Chebyshev distribution. The sample complexity is $O_n(d^{2n}\log d)$, if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most $O(\sigma)$, and the run-time depends on $\log(1/\sigma)$. In the setting where each $\mathbf{x}_i$ and $y_i$ are known up to $N$ bits of precision, the run-time's dependence on $N$ is linear. We also show that our sample complexities are optimal in terms of $d^n$. Furthermore, we show that it is possible to have the run-time be independent of $1/\sigma$, at the cost of a higher sample complexity.

In dimension $d$, Mutually Unbiased Bases (MUBs) are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Constructions to achieve this bound are known when $d$ is some power of prime. The situation is more restrictive in other cases and also when we consider the results over real rather than complex. Thus, certain relaxations of this model are considered in literature and consequently Approximate MUBs (AMUB) are studied. This enables one to construct potentially large number of such objects for $\mathbb{C}^d$ as well as in $\mathbb{R}^d$. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product $|\braket{v_1|v_2}|$ to be two-valued, one being 0 and the other $ \leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$, such that $\lambda > 0$ and the numerator $1 + \mathcal{O}(d^{-\lambda}) \leq 2$. Each such vector constructed, has an important feature that large number of its components are zero and the non-zero components are of equal magnitude. Our techniques are based on combinatorial structures related to RBDs. We show that for several composite dimensions $d$, one can construct $\mathcal{O}(\sqrt{d})$ many APMUBs, in which cases the number of MUBs are significantly small. To be specific, this result works for $d$ of the form $(q-e)(q+f), \ q, e, f \in \mathbb{N}$, with the conditions $0 \leq f \leq e$ for constant $e, f$ and $q$ some power of prime. We also show that such APMUBs provide sets of Bi-angular vectors which are $\mathcal{O}(d^{\frac{3}{2}})$ in numbers, having high angular distances among them. Finally, as the MUBs are equivalent to a set of Hadamard matrices, we show that the APMUBs are so with the set of Weighing matrices.

In this paper, a rate adaptive geometric constellation shaping (GCS) scheme which is fully backward-compatible with existing state of the art bit-interleaved coded modulation (BICM) systems is proposed and experimentally demonstrated. The system relies on optimization of the positions of the quadrature amplitude modulation (QAM) points on the I/Q plane for maximized achievable information rate, while maintaining quantization and fiber nonlinear noise robustness. Furthermore, `dummy' bits are multiplexed with coded bits before mapping to symbols. Rate adaptivity is achieved by tuning the ratio of coded and `dummy' bits, while maintaining a fixed forward error-correction block and a fixed modulation format size. The points' positions and their labeling are optimized using automatic differentiation. The proposed GCS scheme is compared to a time-sharing hybrid (TH) QAM modulation and the now mainstream probabilistic amplitude shaping (PAS) scheme. The TH without shaping is outperformed for all studied data rates in a simulated linear channel by up to 0.7 dB. In a linear channel, PAS is shown to outperform the proposed GCS scheme, while similar performances are reported for PAS and the proposed GCS in a simulated nonlinear fiber channel. The GCS scheme is experimentally demonstrated in a multi-span recirculating loop coherent optical fiber transmission system with a total distance of up to 3000 km. Near-continuous zero-error flexible throughput is reported as a function of the transmission distance. Up to 1-2 spans of increased reach gains are achieved at the same net data rate w.r.t. conventional QAM. At a given distance, up to 0.79 bits/2D symbol of gain w.r.t. conventional QAM is achieved. In the experiment, similar performance to PAS is demonstrated.

For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for ${\sf M}$ over the poset $P$. For $2$-parameter persistence modules, recently a zigzag persistence based algorithm has been proposed that takes advantage of the fact that generalized rank for $2$-parameter modules is equal to the number of full intervals in a zigzag module defined on the boundary of the poset. Analogous definition of boundary for $d$-parameter persistence modules or general $P$-indexed persistence modules does not seem plausible. To overcome this difficulty, we first unfold a given $P$-indexed module ${\sf M}$ into a zigzag module ${\sf M}_{ZZ}$ and then check how many full interval modules in a decomposition of ${\sf M}_{ZZ}$ can be folded back to remain full in ${\sf M}$. This number determines the generalized rank of ${\sf M}$. For special cases of degree-$d$ homology for $d$-complexes, we obtain a more efficient algorithm including a linear time algorithm for degree-$1$ homology in graphs.

Let $X$ and $Z$ be random vectors, and $Y=g(X,Z)$. In this paper, on the one hand, for the case that $X$ and $Z$ are continuous, by using the ideas from the total variation and the flux of $g$, we develop a point of view in causal inference capable of dealing with a broad domain of causal problems. Indeed, we focus on a function, called Probabilistic Easy Variational Causal Effect (PEACE), which can measure the direct causal effect of $X$ on $Y$ with respect to continuously and interventionally changing the values of $X$ while keeping the value of $Z$ constant. PEACE is a function of $d\ge 0$, which is a degree managing the strengths of probability density values $f(x|z)$. On the other hand, we generalize the above idea for the discrete case and show its compatibility with the continuous case. Further, we investigate some properties of PEACE using measure theoretical concepts. Furthermore, we provide some identifiability criteria and several examples showing the generic capability of PEACE. We note that PEACE can deal with the causal problems for which micro-level or just macro-level changes in the value of the input variables are important. Finally, PEACE is stable under small changes in $\partial g_{in}/\partial x$ and the joint distribution of $X$ and $Z$, where $g_{in}$ is obtained from $g$ by removing all functional relationships defining $X$ and $Z$.

In this paper we develop a novel nonparametric framework to test the independence of two random variables $\mathbf{X}$ and $\mathbf{Y}$ with unknown respective marginals $H(dx)$ and $G(dy)$ and joint distribution $F(dx dy)$, based on {\it Receiver Operating Characteristic} (ROC) analysis and bipartite ranking. The rationale behind our approach relies on the fact that, the independence hypothesis $\mathcal{H}\_0$ is necessarily false as soon as the optimal scoring function related to the pair of distributions $(H\otimes G,\; F)$, obtained from a bipartite ranking algorithm, has a ROC curve that deviates from the main diagonal of the unit square.We consider a wide class of rank statistics encompassing many ways of deviating from the diagonal in the ROC space to build tests of independence. Beyond its great flexibility, this new method has theoretical properties that far surpass those of its competitors. Nonasymptotic bounds for the two types of testing errors are established. From an empirical perspective, the novel procedure we promote in this paper exhibits a remarkable ability to detect small departures, of various types, from the null assumption $\mathcal{H}_0$, even in high dimension, as supported by the numerical experiments presented here.

With the proliferation of spatio-textual data, Top-k KNN spatial keyword queries (TkQs), which return a list of objects based on a ranking function that evaluates both spatial and textual relevance, have found many real-life applications. Existing geo-textual indexes for TkQs use traditional retrieval models like BM25 to compute text relevance and usually exploit a simple linear function to compute spatial relevance, but its effectiveness is limited. To improve effectiveness, several deep learning models have recently been proposed, but they suffer severe efficiency issues. To the best of our knowledge, there are no efficient indexes specifically designed to accelerate the top-k search process for these deep learning models. To tackle these issues, we propose a novel technique, which Learns to Index the Spatio-Textual data for answering embedding based spatial keyword queries (called LIST). LIST is featured with two novel components. Firstly, we propose a lightweight and effective relevance model that is capable of learning both textual and spatial relevance. Secondly, we introduce a novel machine learning based Approximate Nearest Neighbor Search (ANNS) index, which utilizes a new learning-to-cluster technique to group relevant queries and objects together while separating irrelevant queries and objects. Two key challenges in building an effective and efficient index are the absence of high-quality labels and unbalanced clustering results. We develop a novel pseudo-label generation technique to address the two challenges. Experimental results show that LIST significantly outperforms state-of-the-art methods on effectiveness, with improvements up to 19.21% and 12.79% in terms of NDCG@1 and Recall@10, and is three orders of magnitude faster than the most effective baseline.

Quantum circuit compilation comprises many computationally hard reasoning tasks that nonetheless lie inside #$\mathbf{P}$ and its decision counterpart in $\mathbf{PP}$. The classical simulation of general quantum circuits is a core example. We show for the first time that a strong simulation of universal quantum circuits can be efficiently tackled through weighted model counting by providing a linear encoding of Clifford+T circuits. To achieve this, we exploit the stabilizer formalism by Knill, Gottesmann, and Aaronson and the fact that stabilizer states form a basis for density operators. With an open-source simulator implementation, we demonstrate empirically that model counting often outperforms state-of-the-art simulation techniques based on the ZX calculus and decision diagrams. Our work paves the way to apply the existing array of powerful classical reasoning tools to realize efficient quantum circuit compilation; one of the obstacles on the road towards quantum supremacy.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.