Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these manifolds, given their intrinsic dimensionality, volume and curvature. However, such works employ definitions of volume and curvature that are inherently difficult to compute. Therefore such theory cannot be easily tested against numerical simulations to understand the tightness of the proven bounds. We instead study typical distortions arising in random projections of an ensemble of smooth Gaussian random manifolds. We find explicitly computable, approximate theoretical bounds on the number of projections required to accurately preserve the geometry of these manifolds. Our bounds, while approximate, can only be violated with a probability that is exponentially small in the ambient dimension, and therefore they hold with high probability in cases of practical interest. Moreover, unlike previous work, we test our theoretical bounds against numerical experiments on the actual geometric distortions that typically occur for random projections of random smooth manifolds. We find our bounds are tighter than previous results by several orders of magnitude.
Inner Mongolia University（内蒙古大学）；Hainan University（海南大学）；Zhijiang College, Zhejiang University of Technology（浙江工业大学之江学院）；China Agriculture University（中国农业大学）
Cross-manifold clustering is a hard topic and many traditional clustering methods fail because of the cross-manifold structures. In this paper, we propose a Multiple Flat Projections Clustering (MFPC) to deal with cross-manifold clustering problems. In our MFPC, the given samples are projected into multiple subspaces to discover the global structures of the implicit manifolds. Thus, the cross-manifold clusters are distinguished from the various projections. Further, our MFPC is extended to nonlinear manifold clustering via kernel tricks to deal with more complex cross-manifold clustering. A se…
Tutte Institute for Mathematics and Computing（塔特数学与计算研究所）
UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique…
Weifang University（潍坊学院）；The University of Hong Kong（香港大学）；Southwestern University of Finance and Economics（西南财经大学）
In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by the alternating projections onto the tangent spaces of the fixed rank matrices manifo…
This is a tutorial and survey paper on the Johnson-Lindenstrauss (JL) lemma and linear and nonlinear random projections. We start with linear random projection and then justify its correctness by JL lemma and its proof. Then, sparse random projections with $\ell_1$ norm and interpolation norm are introduced. Two main applications of random projection, which are low-rank matrix approximation and approximate nearest neighbor search by random projection onto hypercube, are explained. Random Fourier Features (RFF) and Random Kitchen Sinks (RKS) are explained as methods for nonlinear random project…
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and review earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. We then compute asymptotic equivalents of the generalization performance (in terms of squared bias and variance) of the minimum norm least-squares fit with random projectio…
Random projections are able to perform dimension reduction efficiently for datasets with nonlinear low-dimensional structures. One well-known example is that random matrices embed sparse vectors into a low-dimensional subspace nearly isometrically, known as the restricted isometric property in compressed sensing. In this paper, we explore some applications of random projections in deep neural networks. We provide the expressive power of fully connected neural networks when the input data are sparse vectors or form a low-dimensional smooth manifold. We prove that the number of neurons required …
The approximation of nonlinear kernels via linear feature maps has recently gained interest due to their applications in reducing the training and testing time of kernel-based learning algorithms. Current random projection methods avoid the curse of dimensionality by embedding the nonlinear feature space into a low dimensional Euclidean space to create nonlinear kernels. We introduce a Layered Random Projection (LaRP) framework, where we model the linear kernels and nonlinearity separately for increased training efficiency. The proposed LaRP framework was assessed using the MNIST hand-written …
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Minnesota（明尼苏达大学）
Random projection is often used to project higher-dimensional vectors onto a lower-dimensional space, while approximately preserving their pairwise distances. It has emerged as a powerful tool in various data processing tasks and has attracted considerable research interest. Partly motivated by the recent discoveries in neuroscience, in this paper we study the problem of random projection using binary matrices with controllable sparsity patterns. Specifically, we proposed two sparse binary projection models that work on general data vectors. Compared with the conventional random projection mod…
United States Census Bureau, Center for Economic Studies（美国人口普查局经济研究中心）
The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
In the past few years, Generative Adversarial Networks (GANs) have dramatically advanced our ability to represent and parameterize high-dimensional, non-linear image manifolds. As a result, they have been widely adopted across a variety of applications, ranging from challenging inverse problems like image completion, to problems such as anomaly detection and adversarial defense. A recurring theme in many of these applications is the notion of projecting an image observation onto the manifold that is inferred by the generator. In this context, Projected Gradient Descent (PGD) has been the most …
Purdue University（普渡大学）；Université de Montréal（蒙特利尔大学）
We introduce a novel random projection technique for efficiently reducing the dimension of very high-dimensional tensors. Building upon classical results on Gaussian random projections and Johnson-Lindenstrauss transforms~(JLT), we propose two tensorized random projection maps relying on the tensor train~(TT) and CP decomposition format, respectively. The two maps offer very low memory requirements and can be applied efficiently when the inputs are low rank tensors given in the CP or TT format. Our theoretical analysis shows that the dense Gaussian matrix in JLT can be replaced by a low-rank t…
Chalmers University of Technology（查尔姆斯理工大学）；ENS（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL University（巴黎文理研究大学）；Inria（法国国家信息与自动化研究所）；National University of Singapore（新加坡国立大学）
We propose a generic, Bayesian, information geometric approach to the exploration--exploitation trade-off in multi-armed bandit problems. Our approach, BelMan, uniformly supports pure exploration, exploration--exploitation, and two-phase bandit problems. The knowledge on bandit arms and their reward distributions is summarised by the barycentre of the joint distributions of beliefs and rewards of the arms, the \emph{pseudobelief-reward}, within the beliefs-rewards manifold. BelMan alternates \emph{information projection} and \emph{reverse information projection}, i.e., projection of the pseudo…
We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a lower-dimensional Euclidean space using random projections. Subsequently, estimation proceeds using cyclic monotonicity moment inequalities implied by the multinomial choice model; the estimation procedure is semi-parametric and does not require explicit distributional assumptions to be made regarding the random utility errors. The random projection procedure is …
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In classification problems there are two general techniques using random projections. The first involves many projections in an ensemble -- the idea here is to aggregate the results after applying different random projections, with the aim of achieving superior statistical accuracy. The second class of methods include hashing and sketching techniques, which are str…
Accelerating Birkhoff Projection for Manifold-Constrained Hyper-Connections
School of Statistics（统计学系）；Renmin University of China（中国人民大学）；School of Statistics and Data Science（统计学与数据科学学院）；Institute of Big Data Research（大数据研究院）；Shanghai University of Finance and Economics（上海财经大学）
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Dali University（大理大学）
K-nearest neighbor (kNN) search has wide applications in many areas, including data mining, machine learning, statistics and many applied domains. Inspired by the success of ensemble methods and the flexibility of tree-based methodology, we propose random projection forests (rpForests), for kNN search. rpForests finds kNNs by aggregating results from an ensemble of random projection trees with each constructed recursively through a series of carefully chosen random projections. rpForests achieves a remarkable accuracy in terms of fast decay in the missing rate of kNNs and that of discrepancy i…
Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when project high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random matrix affect norms of random vectors. In particular we prove the distribution of the norm of random vector $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{σ^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1…
Zhejiang University（浙江大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We propose and study kernel conjugate gradient methods (KCGM) with random projections for least-squares regression over a separable Hilbert space. Considering two types of random projections generated by randomized sketches and Nyström subsampling, we prove optimal statistical results with respect to variants of norms for the algorithms under a suitable stopping rule. Particularly, our results show that if the projection dimension is proportional to the effective dimension of the problem, KCGM with randomized sketches can generalize optimally, while achieving a computational advantage. As a co…
Principal Components Regression (PCR) is a traditional tool for dimension reduction in linear regression that has been both criticized and defended. One concern about PCR is that obtaining the leading principal components tends to be computationally demanding for large data sets. While random projections do not possess the optimality properties of the leading principal subspace, they are computationally appealing and hence have become increasingly popular in recent years. In this paper, we present an analysis showing that for random projections satisfying a Johnson-Lindenstrauss embedding prop…
A common challenge in nonparametric inference is its high computational complexity when data volume is large. In this paper, we develop computationally efficient nonparametric testing by employing a random projection strategy. In the specific kernel ridge regression setup, a simple distance-based test statistic is proposed. Notably, we derive the minimum number of random projections that is sufficient for achieving testing optimality in terms of the minimax rate. An adaptive testing procedure is further established without prior knowledge of regularity. One technical contribution is to establi…
