**Non-linear Iterative Partial Least Squares (NIPALS)**

In statistics, non-linear iterative partial least squares (NIPALS) is an algorithm for computing the first few components in a principal component or partial least squares analysis. For very-high-dimensional datasets, such as those generated in the ‘omics sciences (e.g., genomics, metabolomics) it is usually only necessary to compute the first few principal components. The nonlinear iterative partial least squares (NIPALS) algorithm calculates t1 and p1′ from X. The outer product, t1p1’ can then be subtracted from X leaving the residual matrix E1. This can be then used to calculate subsequent principal components. This results in a dramatic reduction in computational time since calculation of the covariance matrix is avoided. … **L-Convex Set**

We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we call ($\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-) extended L-convex sets. We introduce extended L-convex sets as variants of certain discrete structures called L-convex sets and L-convex polyhedra, studied in the field of discrete convex analysis. We also introduce homomorphisms between extended L-convex sets. The theorem claims that there is a one to one correspondence (up to isomorphism) between two classes. The thesis also contains an introductory chapter on enriched categories and no categorical knowledge is assumed. … **SchedNet**

Many real-world reinforcement learning tasks require multiple agents to make sequential decisions under the agents’ interaction, where well-coordinated actions among the agents are crucial to achieve the target goal better at these tasks. One way to accelerate the coordination effect is to enable multiple agents to communicate with each other in a distributed manner and behave as a group. In this paper, we study a practical scenario when (i) the communication bandwidth is limited and (ii) the agents share the communication medium so that only a restricted number of agents are able to simultaneously use the medium, as in the state-of-the-art wireless networking standards. This calls for a certain form of communication scheduling. In that regard, we propose a multi-agent deep reinforcement learning framework, called SchedNet, in which agents learn how to schedule themselves, how to encode the messages, and how to select actions based on received messages. SchedNet is capable of deciding which agents should be entitled to broadcasting their (encoded) messages, by learning the importance of each agent’s partially observed information. We evaluate SchedNet against multiple baselines under two different applications, namely, cooperative communication and navigation, and predator-prey. Our experiments show a non-negligible performance gap between SchedNet and other mechanisms such as the ones without communication and with vanilla scheduling methods, e.g., round robin, ranging from 32% to 43%. … **Variational Deep Embedding (VaDE)**

Clustering is among the most fundamental tasks in computer vision and machine learning. In this paper, we propose Variational Deep Embedding (VaDE), a novel unsupervised generative clustering approach within the framework of Variational Auto-Encoder (VAE). Specifically, VaDE models the data generative procedure with a Gaussian Mixture Model (GMM) and a deep neural network (DNN): 1) the GMM picks a cluster; 2) from which a latent embedding is generated; 3) then the DNN decodes the latent embedding into observables. Inference in VaDE is done in a variational way: a different DNN is used to encode observables to latent embeddings, so that the evidence lower bound (ELBO) can be optimized using Stochastic Gradient Variational Bayes (SGVB) estimator and the reparameterization trick. Quantitative comparisons with strong baselines are included in this paper, and experimental results show that VaDE significantly outperforms the state-of-the-art clustering methods on 4 benchmarks from various modalities. Moreover, by VaDE’s generative nature, we show its capability of generating highly realistic samples for any specified cluster, without using supervised information during training. Lastly, VaDE is a flexible and extensible framework for unsupervised generative clustering, more general mixture models than GMM can be easily plugged in. …

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