Escaping from saddle points and finding local minima is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(\epsilon, \sqrt{\epsilon})$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(\epsilon^{-3.5})$, which is not optimal. In this paper, we propose \texttt{Pullback}, a faster perturbed stochastic gradient framework for finding local minima. We show that Pullback with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(\epsilon, \epsilon_{H})$-approximate local minima within $\tilde O(\epsilon^{-3} + \epsilon_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(\epsilon^{-3})$ when $\epsilon_H = \sqrt{\epsilon}$). The core idea of our framework is a step-size ``pullback'' scheme to control the average movement of the iterates, which leads to faster convergence to the local minima. Experiments on matrix factorization problems corroborate our theory.

Most of the time the equation of the model involves higher-order and higher-degree functions. In this post, we will learn how to solve the harder nonlinear equations using a heuristic algorithm of finding the least-squares approximate solution. Practically, almost all problems can be represented as a set of nonlinear equations. However, solving them is a little more complex than the method we used to apply for solving linear equations. Let us consider a set of \(m \) nonlinear equations and \(n \) variables.

We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the co-occurrences of signed trees for a family of non-isomorphic trees. When the two networks are Erd\H{o}s-R\'enyi random graphs $\mathcal{G}(n,q)$ that are either independent or correlated with correlation coefficient $\rho$, our test runs in $n^{2+o(1)}$ time and succeeds with high probability as $n\to\infty$, provided that $n\min\{q,1-q\} \ge n^{-o(1)}$ and $\rho^2>\alpha \approx 0.338$, where $\alpha$ is Otter's constant so that the number of unlabeled trees with $K$ edges grows as $(1/\alpha)^K$. This significantly improves the prior work in terms of statistical accuracy, running time, and graph sparsity.

In network analysis, developing a unified theoretical framework that can compare methods under different models is an interesting problem. This paper proposes a partial solution to this problem. We summarize the idea of using separation condition for a standard network and sharp threshold of Erd\"os-R\'enyi random graph to study consistent estimation, compare theoretical error rates and requirements on network sparsity of spectral methods under models that can degenerate to stochastic block model as a four-step criterion SCSTC. Using SCSTC, we find some inconsistent phenomena on separation condition and sharp threshold in community detection. Especially, we find original theoretical results of the SPACL algorithm introduced to estimate network memberships under the mixed membership stochastic blockmodel were sub-optimal. To find the formation mechanism of inconsistencies, we re-establish theoretical convergence rates of this algorithm by applying recent techniques on row-wise eigenvector deviation. The results are further extended to the degree corrected mixed membership model. By comparison, our results enjoy smaller error rates, lesser dependence on the number of communities, weaker requirements on network sparsity, and so forth. Furthermore, separation condition and sharp threshold obtained from our theoretical results match classical results, which shows the usefulness of this criterion on studying consistent estimation.

This study is motivated by the critical challenges in the biopharmaceutical manufacturing, including high complexity, high uncertainty, and very limited process data. Each experiment run is often very expensive. To support the optimal and robust process control, we propose a general green simulation assisted policy gradient (GS-PG) framework for both online and offline learning settings. Basically, to address the key limitations of state-of-art reinforcement learning (RL), such as sample inefficiency and low reliability, we create a mixture likelihood ratio based policy gradient estimation that can leverage on the information from historical experiments conducted under different inputs, including process model coefficients and decision policy parameters. Then, to accelerate the learning of optimal and robust policy, we further propose a variance reduction based sample selection method that allows GS-PG to intelligently select and reuse most relevant historical trajectories. The selection rule automatically updates the samples to be reused during the learning of process mechanisms and the search for optimal policy. Our theoretical and empirical studies demonstrate that the proposed framework can perform better than the state-of-art policy gradient approach and accelerate the optimal robust process control for complex stochastic systems under high uncertainty.

The binomial distribution is a probability distribution that applies to binomial experiments. The binomial distribution may be imagined as the probability distribution of a number of heads that appear on a coin flip in a specific experiment comprising of a fixed number of coin flips. In this blog post, we will learn binomial distribution with the help of examples. If you are an aspiring data scientist looking forward to learning/understand the binomial distribution in a better manner, this post might be very helpful. The binomial distribution is a discrete probability distribution that represents the probabilities of binomial random variables in a binomial experiment.

The set of local modes and the ridge lines estimated from a dataset are important summary characteristics of the data-generating distribution. In this work, we consider estimating the local modes and ridges from point cloud data in a product space with two or more Euclidean/directional metric spaces. Specifically, we generalize the well-known (subspace constrained) mean shift algorithm to the product space setting and illuminate some pitfalls in such generalization. We derive the algorithmic convergence of the proposed method, provide practical guidelines on the implementation, and demonstrate its effectiveness on both simulated and real datasets.

Unfortunately, no one can be told what the matrix is. You have to see yourself. In this blog, lets continue the topic from previous blog where we left off. Blog covers 3-D Linear transformations, determinants, Inverse Matrices, Column Space, Null Space, Non Square matrices as transformations between dimensions. Example: Consider a linear transformation with 3-Dimentional vectors as inputs and 3-Dimentional vectors as output.

In the present work, we describe a framework for modeling how models can be built that integrates concepts and methods from a wide range of fields. The information schism between the real-world and that which can be gathered and considered by any individual information processing agent is characterized and discussed, followed by the presentation of a series of the adopted requisites while developing the modeling approach. The issue of mapping from datasets into models is subsequently addressed, as well as some of the respectively implied difficulties and limitations. Based on these considerations, an approach to meta modeling how models are built is then progressively developed. First, the reference M* meta model framework is presented, which relies critically in associating whole datasets and respective models in terms of a strict equivalence relation. Among the interesting features of this model are its ability to bridge the gap between data and modeling, as well as paving the way to an algebra of both data and models which can be employed to combine models into hierarchical manner. After illustrating the M* model in terms of patterns derived from regular lattices, the reported modeling approach continues by discussing how sampling issues, error and overlooked data can be addressed, leading to the $M^{<\epsilon>}$ variant, illustrated respectively to number theory. The situation in which the data needs to be represented in terms of respective probability densities is treated next, yielding the $M^{<\sigma>}$ meta model, which is then illustrated respectively to a real-world dataset (iris flowers data). Several considerations about how the developed framework can provide insights about data clustering, complexity, collaborative research, deep learning, and creativity are then presented, followed by overall conclusions.

This paper deals with the problem of graph matching or network alignment for Erd\H{o}s--R\'enyi graphs, which can be viewed as a noisy average-case version of the graph isomorphism problem. Let $G$ and $G'$ be $G(n, p)$ Erd\H{o}s--R\'enyi graphs marginally, identified with their adjacency matrices. Assume that $G$ and $G'$ are correlated such that $\mathbb{E}[G_{ij} G'_{ij}] = p(1-\alpha)$. For a permutation $\pi$ representing a latent matching between the vertices of $G$ and $G'$, denote by $G^\pi$ the graph obtained from permuting the vertices of $G$ by $\pi$. Observing $G^\pi$ and $G'$, we aim to recover the matching $\pi$. In this work, we show that for every $\varepsilon \in (0,1]$, there is $n_0>0$ depending on $\varepsilon$ and absolute constants $\alpha_0, R > 0$ with the following property. Let $n \ge n_0$, $(1+\varepsilon) \log n \le np \le n^{\frac{1}{R \log \log n}}$, and $0 < \alpha < \min(\alpha_0,\varepsilon/4)$. There is a polynomial-time algorithm $F$ such that $\mathbb{P}\{F(G^\pi,G')=\pi\}=1-o(1)$. This is the first polynomial-time algorithm that recovers the exact matching between vertices of correlated Erd\H{o}s--R\'enyi graphs with constant correlation with high probability. The algorithm is based on comparison of partition trees associated with the graph vertices.