For instance, one would expect certain levels of similarities between the We consider the problem of estimating the topology of multiple networks brain networks of different healthy individuals or between the same from nodal observations, where these networks are assumed social network observed at different points in time. Prominent methods to be drawn from the same (unknown) random graph model. We for multiple network inference include statistical approaches, adopt a graphon as our random graph model, which is a nonparametric primarily consisting of the joint estimation of Gaussian graphical model from which graphs of potentially different sizes can models [13-17]. These methods typically involve modifications on be drawn. The versatility of graphons allows us to tackle the joint the graphical lasso formulation with additional encouragement of inference problem even for the cases where the graphs to be recovered structural similarity. Estimation of time-varying graphs is widely contain different number of nodes and lack precise alignment popular, as the relationship between graphs is typically straightforward across the graphs. Our solution is based on combining a maximum to implement by considering that graph variation is smooth likelihood penalty with graphon estimation schemes and can be used across time [18, 19]. The above methods for estimating multiple networks to augment existing network inference methods. We validate our typically enforce similar structure, such as promoting similar proposed approach by comparing its performance against competing sparsity patterns [20].

We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost for big data. In this paper, we propose a new Bayesian approach, EigenGP, that learns both basis dictionary elements -- eigenfunctions of a GP prior -- and prior precisions in a sparse finite model. It is well known that, among all orthogonal basis functions, eigenfunctions can provide the most compact representation. Unlike other sparse Bayesian finite models where the basis function has a fixed form, our eigenfunctions live in a reproducing kernel Hilbert space as a finite linear combination of kernel functions. We learn the dictionary elements -- eigenfunctions -- and the prior precisions over these elements as well as all the other hyperparameters from data by maximizing the model marginal likelihood. We explore computational linear algebra to simplify the gradient computation significantly. Our experimental results demonstrate improved predictive performance of EigenGP over alternative sparse GP methods as well as relevance vector machines.

Compilation techniques in planning reformulate a problem into an alternative encoding for which efficient, off-the-shelf solvers are available. In this work, we present a novel mixed-integer linear programming (MILP) compilation for cost-optimal numeric planning with instantaneous actions. While recent works on the problem are restricted to actions that modify variables present in simple numeric conditions, our MILP formulation, in addition, handles linear conditions and linear action effects on numeric state variables. Such problems are particularly challenging due to the state-dependency of the action effects. Experiments show that our approach, in addition to being the state of the art for the more general problem class, is competitive with heuristic search-based planners on domains with only simple numeric conditions.

A live interactive narrative (LIN) is an experience where multiple players take on fictional roles and interact with real-world objects and actors to participate in a pre-authored narrative. Temporal properties of LINs are important to its viability and aesthetic quality and hence deserve special design consideration. In this paper, we tackle the largely overlooked problem of scheduling a multiplayer interactive narrative and propose the Live Interactive Narrative Scheduling Problem (LINSP), which handles reasoning under temporal uncertainty, resource scheduling, and non-linear plot choices. We present a mixed-integer linear programming formulation of the problem and empirically evaluates its scalability over large narrative instances.

Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of its adjacency or Laplacian matrix, and has found applications throughout the sciences. Many such networks are multipartite, meaning their nodes can be divided into partitions and nodes of the same partition are never connected. When the network is multipartite, this paper demonstrates that the node representations obtained via spectral embedding live near partition-specific low-dimensional subspaces of a higher-dimensional ambient space. For this reason we propose a follow-on step after spectral embedding, to recover node representations in their intrinsic rather than ambient dimension, proving uniform consistency under a low-rank, inhomogeneous random graph model. Our method naturally generalizes bipartite spectral embedding, in which node representations are obtained by singular value decomposition of the biadjacency or bi-Laplacian matrix.

Probability for improved business decisions: Introduction, Combinatorics, Bayesian Inference, Distributions. Welcome to Probability and Statistics for Business and Data Science! In this course we cover what you need to know about probability and statistics to succeed in business and the data science field! This practical course will go over theory and implementation of statistics to real world problems. Each section has example problems, in course quizzes, and assessment tests.

We introduce a simple and fast method for comparing graphs of different sizes. Existing approaches are often either limited to comparing graphs with the same number of vertices or are computationally unscalable. We propose the Embedded Laplacian Distance (ELD) for comparing graphs of potentially vastly different sizes. Our approach first projects the graphs onto a common, low-dimensional Laplacian embedding space that respects graphical structure. This reduces the problem to that of comparing point clouds in a Euclidean space. A distance can then be computed efficiently via a natural sliced Wasserstein approach. We show that the ELD is a pseudo-metric and is invariant under graph isomorphism. We provide intuitive interpretations of the ELD using tools from spectral graph theory. We test the efficacy of the ELD approach extensively on both simulated and real data. Results obtained are excellent.

We address the following foundational question: what is the population, and sample, Frechet mean (or median) graph of an ensemble of inhomogeneous Erdos-Renyi random graphs? We prove that if we use the Hamming distance to compute distances between graphs, then the Frechet mean (or median) graph of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix of the ensemble. We show that the result also holds for the sample mean (or median) when the population expected adjacency matrix is replaced with the sample mean adjacency matrix. Consequently, the Frechet mean (or median) graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold: it is either the empty graph, or the complete graph. This novel theoretical result has some significant practical consequences; for instance, the Frechet mean of an ensemble of sparse inhomogeneous random graphs is always the empty graph.

Newgen Software, a leading provider of a unified digital transformation platform, is pleased to announce that it is acquiring India-based Number Theory, an AI/ML (artificial intelligence and machine learning) data science platform company, subject to the completion of conditions as stated in the approved Share Purchase Agreement. Number Theory's platform, AI Studio, brings intuitive AI/ML to every enterprise, while unifying the entire lifecycle of data engineering, from data preparation to model development and monitoring. It empowers both citizen and expert data scientists to work faster and more efficiently, thereby helping in accomplishing key machine learning tasks in just hours or days, not months. This acquisition will further strengthen Newgen's low code digital transformation platform, NewgenONE, with AI/ML modeling and data analytics capabilities. "Our customers are increasingly looking to leverage data for deeper insights and accelerated growth. Number Theory will bring domain expertise, along with a powerful engine to extract actionable insights in real time. AI/ML projects often get complex, expensive, and not rewarding. What we like about Number Theory's platform is that it is for every enterprise. It lets fusion teams build, deploy, and collaborate on the entire modeling lifecycle in low code and on cloud. We look forward to welcoming the Number Theory team to the Newgen family," said Virender Jeet, CEO, Newgen.