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We introduce a novel sufficient dimension-reduction (SDR) method which is robust against outliers using $\alpha$-distance covariance (dCov) in dimension-reduction problems. Under very mild conditions on the predictors, the central subspace is effectively estimated and model-free advantage without estimating link function based on the projection on the Stiefel manifold. We establish the convergence property of the proposed estimation under some regularity conditions. We compare the performance of our method with existing SDR methods by simulation and real data analysis and show that our algorithm improves the computational efficiency and effectiveness.

Identification and analysis of symmetrical patterns in the natural world have led to significant discoveries across various scientific fields, such as the formulation of gravitational laws in physics and advancements in the study of chemical structures. In this paper, we focus on exploiting Euclidean symmetries inherent in certain cooperative multi-agent reinforcement learning (MARL) problems and prevalent in many applications. We begin by formally characterizing a subclass of Markov games with a general notion of symmetries that admits the existence of symmetric optimal values and policies. Motivated by these properties, we design neural network architectures with symmetric constraints embedded as an inductive bias for multi-agent actor-critic methods. This inductive bias results in superior performance in various cooperative MARL benchmarks and impressive generalization capabilities such as zero-shot learning and transfer learning in unseen scenarios with repeated symmetric patterns. The code is available at: //github.com/dchen48/E3AC.

Maximizing the log-likelihood is a crucial aspect of learning latent variable models, and variational inference (VI) stands as the commonly adopted method. However, VI can encounter challenges in achieving a high log-likelihood when dealing with complicated posterior distributions. In response to this limitation, we introduce a novel variational importance sampling (VIS) approach that directly estimates and maximizes the log-likelihood. VIS leverages the optimal proposal distribution, achieved by minimizing the forward $\chi^2$ divergence, to enhance log-likelihood estimation. We apply VIS to various popular latent variable models, including mixture models, variational auto-encoders, and partially observable generalized linear models. Results demonstrate that our approach consistently outperforms state-of-the-art baselines, both in terms of log-likelihood and model parameter estimation.

We give an alternative derivation of $(N,N)$-isogenies between fastKummer surfaces which complements existing works based on the theory oftheta functions. We use this framework to produce explicit formulae for thecase of $N = 3$, and show that the resulting algorithms are more efficient thanall prior $(3, 3)$-isogeny algorithms.

Maximizing a non-negative, monontone, submodular function $f$ over $n$ elements under a cardinality constraint $k$ (SMCC) is a well-studied NP-hard problem. It has important applications in, e.g., machine learning and influence maximization. Though the theoretical problem admits polynomial-time approximation algorithms, solving it in practice often involves frequently querying submodular functions that are expensive to compute. This has motivated significant research into designing parallel approximation algorithms in the adaptive complexity model; adaptive complexity (adaptivity) measures the number of sequential rounds of $\text{poly}(n)$ function queries an algorithm requires. The state-of-the-art algorithms can achieve $(1-\frac{1}{e}-\varepsilon)$-approximate solutions with $O(\frac{1}{\varepsilon^2}\log n)$ adaptivity, which approaches the known adaptivity lower-bounds. However, the $O(\frac{1}{\varepsilon^2} \log n)$ adaptivity only applies to maximizing worst-case functions that are unlikely to appear in practice. Thus, in this paper, we consider the special class of $p$-superseparable submodular functions, which places a reasonable constraint on $f$, based on the parameter $p$, and is more amenable to maximization, while also having real-world applicability. Our main contribution is the algorithm LS+GS, a finer-grained version of the existing LS+PGB algorithm, designed for instances of SMCC when $f$ is $p$-superseparable; it achieves an expected $(1-\frac{1}{e}-\varepsilon)$-approximate solution with $O(\frac{1}{\varepsilon^2}\log(p k))$ adaptivity independent of $n$. Additionally, unrelated to $p$-superseparability, our LS+GS algorithm uses only $O(\frac{n}{\varepsilon} + \frac{\log n}{\varepsilon^2})$ oracle queries, which has an improved dependence on $\varepsilon^{-1}$ over the state-of-the-art LS+PGB; this is achieved through the design of a novel thresholding subroutine.

Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this paper we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We establish the identifiability of our model and propose a computationally efficient estimation procedure based on the higher-order orthogonal iteration algorithm (HOOI) for tensor SVD composed with a simplex corner-finding algorithm. We then demonstrate the consistency of our estimation procedure by providing a per-node error bound, which showcases the effect of higher-order structures on estimation accuracy. To prove our consistency result, we develop the $\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent, heteroskedastic, subgaussian noise that may be of independent interest. Our analysis uses a novel leave-one-out construction for the iterates, and our bounds depend only on spectral properties of the underlying low-rank tensor under nearly optimal signal-to-noise ratio conditions such that tensor SVD is computationally feasible. Finally, we apply our methodology to real and simulated data, demonstrating some effects not identifiable from the model with discrete community memberships.

Although adaptive gradient methods have been extensively used in deep learning, their convergence rates have not been thoroughly studied, particularly with respect to their dependence on the dimension. This paper considers the classical RMSProp and its momentum extension and establishes the convergence rate of $\frac{1}{T}\sum_{k=1}^TE\left[\|\nabla f(x^k)\|_1\right]\leq O(\frac{\sqrt{d}}{T^{1/4}})$ measured by $\ell_1$ norm without the bounded gradient assumption, where $d$ is the dimension of the optimization variable and $T$ is the iteration number. Since $\|x\|_2\ll\|x\|_1\leq\sqrt{d}\|x\|_2$ for problems with extremely large $d$, our convergence rate can be considered to be analogous to the $\frac{1}{T}\sum_{k=1}^TE\left[\|\nabla f(x^k)\|_2\right]\leq O(\frac{1}{T^{1/4}})$ one of SGD measured by $\ell_1$ norm.

Time Complexity is an important metric to compare algorithms based on their cardinality. The commonly used, trivial notations to qualify the same are the Big-Oh, Big-Omega, Big-Theta, Small-Oh, and Small-Omega Notations. All of them, consider time a part of the real entity, i.e., Time coincides with the horizontal axis in the argand plane. But what if the Time rather than completely coinciding with the real axis of the argand plane, makes some angle with it? We are trying to focus on the case when the Time Complexity will have both real and imaginary components. For Instance, if $T\left(n\right)=\ n\log{n}$, the existing asymptomatic notations are capable of handling that in real time But, if we come across a problem where, $T\left(n\right)=\ n\log{n}+i\cdot n^2$, where, $i=\sqrt[2]{-1}$, the existing asymptomatic notations will not be able to catch up. To mitigate the same, in this research, we would consider proposing the Zeta Notation ($\zeta$), which would qualify Time in both the Real and Imaginary Axis, as per the Argand Plane.

We consider the problem of finding a geodesic disc of smallest radius containing at least $k$ points from a set of $n$ points in a simple polygon that has $m$ vertices, $r$ of which are reflex vertices. We refer to such a disc as a SKEG disc. We present an algorithm to compute a SKEG disc using higher-order geodesic Voronoi diagrams with worst-case time $O(k^{2} n + k^{2} r + \min(kr, r(n-k)) + m)$ ignoring polylogarithmic factors. We then present two $2$-approximation algorithms that find a geodesic disc containing at least $k$ points whose radius is at most twice that of a SKEG disc. The first algorithm computes a $2$-approximation with high probability in $O((n^{2} / k) \log n \log r + m)$ worst-case time with $O(n + m)$ space. The second algorithm runs in $O(n \log^{2} n \log r + m)$ expected time using $O(n + m)$ expected space, independent of $k$. Note that the first algorithm is faster when $k \in \omega(n / \log n)$.

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tilde{\Omega}(\sqrt{\log n})$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.