Label space dimension reduction (LSDR) is an efficient and effective paradigm for multi-label classification with many classes. Existing approaches to LSDR, such as compressive sensing and principal label space transformation, exploit only the label part of the dataset, but not the feature part. In this paper, we propose a novel approach to LSDR that considers both the label and the feature parts. The approach, called conditional principal label space transformation, is based on minimizing an upper bound of the popular Hamming loss. The minimization step of the approach can be carried out efficiently by a simple use of singular value decomposition.

Developing interpretable machine learning models has become an increasingly important issue. One way in which data scientists have been able to develop interpretable models has been to use dimension reduction techniques. In this paper, we examine several dimension reduction techniques including two recent approaches developed in the network psychometrics literature called exploratory graph analysis (EGA) and unique variable analysis (UVA). We compared EGA and UVA with two other dimension reduction techniques common in the machine learning literature (principal component analysis and independent component analysis) as well as no reduction to the variables real data. We show that EGA and UVA perform as well as the other reduction techniques or no reduction. Consistent with previous literature, we show that dimension reduction can decrease, increase, or provide the same accuracy as no reduction of variables. Our tentative results find that dimension reduction tends to lead to better performance when used for classification tasks.

We address the problem of sufficient dimension reduction for feature matrices, which arises often in sensor network localization, brain neuroimaging, and electroencephalography analysis. In general, feature matrices have both row- and column-wise interpretations and contain structural information that can be lost with naive vectorization approaches. To address this, we propose a method called principal support matrix machine (PSMM) for the matrix sufficient dimension reduction. The PSMM converts the sufficient dimension reduction problem into a series of classification problems by dividing the response variables into slices. It effectively utilizes the matrix structure by finding hyperplanes with rank-1 normal matrix that optimally separate the sliced responses. Additionally, we extend our approach to the higher-order tensor case. Our numerical analysis demonstrates that the PSMM outperforms existing methods and has strong interpretability in real data applications.

We provide a decision theoretic analysis of bandit experiments. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for these experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distributions of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and thereby suggests a practical strategy for dimension reduction. The PDEs characterizing minimal Bayes risk can be solved efficiently using sparse matrix routines. We derive the optimal Bayes and minimax policies from their numerical solutions. These optimal policies substantially dominate existing methods such as Thompson sampling and UCB, often by a factor of two. The framework also covers time discounting and pure exploration.

Quantum computers are believed to have the ability to process huge data sizes which can be seen in machine learning applications. In these applications, the data in general is classical. Therefore, to process them on a quantum computer, there is a need for efficient methods which can be used to map classical data on quantum states in a concise manner. On the other hand, to verify the results of quantum computers and study quantum algorithms, we need to be able to approximate quantum operations into forms that are easier to simulate on classical computers with some errors. Motivated by these needs, in this paper we study the approximation of matrices and vectors by using their tensor products obtained through successive Schmidt decompositions. We show that data with distributions such as uniform, Poisson, exponential, or similar to these distributions can be approximated by using only a few terms which can be easily mapped onto quantum circuits. The examples include random data with different distributions, the Gram matrices of iris flower, handwritten digits, 20newsgroup, and labeled faces in the wild. And similarly, some quantum operations such as quantum Fourier transform and variational quantum circuits with a small depth also may be approximated with a few terms that are easier to simulate on classical computers. Furthermore, we show how the method can be used to simplify quantum Hamiltonians: In particular, we show the application to randomly generated transverse field Ising model Hamiltonians. The reduced Hamiltonians can be mapped into quantum circuits easily and therefore can be simulated more efficiently.

The rise in data has led to the need for dimension reduction techniques, especially in the area of non-scalar variables, including time series, natural language processing, and computer vision. In this paper, we specifically investigate dimension reduction for time series through functional data analysis. Current methods for dimension reduction in functional data are functional principal component analysis and functional autoencoders, which are limited to linear mappings or scalar representations for the time series, which is inefficient. In real data applications, the nature of the data is much more complex. We propose a non-linear function-on-function approach, which consists of a functional encoder and a functional decoder, that uses continuous hidden layers consisting of continuous neurons to learn the structure inherent in functional data, which addresses the aforementioned concerns in the existing approaches. Our approach gives a low dimension latent representation by reducing the number of functional features as well as the timepoints at which the functions are observed. The effectiveness of the proposed model is demonstrated through multiple simulations and real data examples.

This course introduces you to one of the main types of Machine Learning: Unsupervised Learning. You will learn how to find insights from data sets that do not have a target or labeled variable. You will learn several clustering and dimension reduction algorithms for unsupervised learning as well as how to select the algorithm that best suits your data. The hands-on section of this course focuses on using best practices for unsupervised learning. By the end of this course you should be able to: Explain the kinds of problems suitable for Unsupervised Learning approaches Explain the curse of dimensionality, and how it makes clustering difficult with many features Describe and use common clustering and dimensionality-reduction algorithms Try clustering points where appropriate, compare the performance of per-cluster models Understand metrics relevant for characterizing clusters Who should take this course?

We consider the problem of parameter estimation under local information constraints, where the estimation algorithm has access to only limited information about each sample. These constraints can be of various types, including communication constraints, where each sample must be described using a few (e.g., constant number of) bits; (local) privacy constraints, where each sample is obtained from a different user and the users seek to reveal as little as possible about their specific data; as well as many others, e.g., noisy communication channels, or limited types of data access such as linear measurements. Such problems have received a lot of attention in recent years, motivated by applications such as data analytics in distributed systems and federated learning. Our main focus is on information-theoretic lower bounds for the minimax error rates (or, equivalently, the sample complexity) of these problems. Several recent works have provided different bounds that apply to specific constraints or work for specific parametric estimation problems, sometimes without allowing for interactive protocols. Indeed, handling interactive protocols is technically challenging, and several results in prior work exhibit flaws in their analysis. In particular, even the most basic Gaussian mean estimation problem using interactive communication remains, quite surprisingly, open. We present general, "plug-and-play" lower bounds for parametric estimation under information constraints that can be used for any local information constraint and allows for interactive protocols. Our abstract bound requires very simple (and natural) assumptions to hold for the underlying parametric family; in particular, we do not require technical "regularity" conditions that are common in asymptotic statistics.

Sufficient dimension reduction is a powerful tool to extract core information hidden in the high-dimensional data and has potentially many important applications in machine learning tasks. However, the existing nonlinear sufficient dimension reduction methods often lack the scalability necessary for dealing with large-scale data. We propose a new type of stochastic neural network under a rigorous probabilistic framework and show that it can be used for sufficient dimension reduction for large-scale data. The proposed stochastic neural network is trained using an adaptive stochastic gradient Markov chain Monte Carlo algorithm, whose convergence is rigorously studied in the paper as well. Through extensive experiments on real-world classification and regression problems, we show that the proposed method compares favorably with the existing state-of-the-art sufficient dimension reduction methods and is computationally more efficient for large-scale data.

Recently, deep convolutional neural networks (CNNs) have achieved many eye-catching results. However, deploying CNNs on resource-constrained edge devices is constrained by limited memory bandwidth for transmitting large intermediated data during inference, i.e., activation. Existing research utilizes mixed-precision and dimension reduction to reduce computational complexity but pays less attention to its application for activation compression. To further exploit the redundancy in activation, we propose a learnable mixed-precision and dimension reduction co-design system, which separates channels into groups and allocates specific compression policies according to their importance. In addition, the proposed dynamic searching technique enlarges search space and finds out the optimal bit-width allocation automatically. Our experimental results show that the proposed methods improve 3.54%/1.27% in accuracy and save 0.18/2.02 bits per value over existing mixed-precision methods on ResNet18 and MobileNetv2, respectively.