This paper surveys various methods for dimensionality reduction and nearest neighbor search.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving essential information.
method Linear and nonlinear random projections, including sparse random projections, random Fourier Features, and Random Kitchen Sinks.
result Various methods for dimensionality reduction and nearest neighbor search are explained and compared.
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
Paper studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.
problem Improving computational efficiency and accuracy in random projections.
method Proposes two sparse binary projection models with controllable sparsity patterns.
result Significant computational advantages and improved accuracies in empirical evaluations.
Free Random Projection enhances reinforcement learning by naturally incorporating hierarchical structure.
problem Improving reinforcement learning algorithms for better generalization and adaptability.
method Introduces Free Random Projection, a method that uses free probability theory to create random orthogonal matrices encoding hierarchical structure.
result Empirically shows consistent improvement in generalization over standard methods on multi-environment benchmarks.
Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problem…
Tensorized random projections reduce high-dimensional tensor size efficiently.
problem Efficiently reducing the dimension of very high-dimensional tensors.
method Proposes two tensorized random projection maps using TT and CP decompositions.
result TT format offers superior performance in terms of required random projection size.
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.
Study shows double descent curve in high-dimensional linear regression with random projections.
problem Understanding the generalization performance in high-dimensional settings with random projections.
method Fixed prediction problem, ridge regression estimator, minimum norm least-squares fit, random matrix theory, asymptotic equivalents.
result Exhibit a double descent curve for high-dimensional linear regression with random projections.
A new framework for dimension reduction using ensemble of random projections.
problem High-dimensional regression problems with limited data.
method Aggregating an ensemble of carefully chosen random projections, retaining based on empirical performance, and selecting singular vectors.
result The proposed method stabilizes error as the number of projection groups increases.
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
The paper examines how well node similarities are preserved by random projections in graph embeddings.
problem The preservation of node similarities under random projections in graph embeddings.
method Investigation of dot product and cosine similarity preservation by random projections over graph matrix rows.
result Random projections produce unreliable embeddings for dot product, especially for high-degree nodes.
A new ensemble method using random projections for kNN classification.
problem Improving kNN classification accuracy through ensemble methods.
method Random projection of bootstrap samples into lower dimensions, using extended neighbourhood rule for base learners.
result Enhanced classification accuracy compared to traditional kNN and other ensembles.
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. T…
The paper introduces DP algorithms using random projections and sign random projections for improved privacy in machine learning.
problem Improving differential privacy in machine learning applications.
method Developed algorithms based on random projections and sign random projections, focusing on individual differential privacy (iDP) and standard differential privacy (DP).
result DP-SignOPORP and iDP-SignRP achieve superior performance in differential privacy, especially for small epsilon values.
We develop a probabilistic framework for sequential random projection.
problem Challenges of sequential decision-making under uncertainty.
method Novel construction of a stopped process and method of mixtures.
result Achieved a non-asymptotic probability bound for random projection.
Corrects bias in random sampling matrices for improved ML methods.
problem Inversion bias in random sampling matrices hampers ML applications.
method Corrects inversion bias for various random sampling methods.
result Establishes local convergence rates for sub-sampled Newton methods.
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 …
Tensorized Rademacher projections outperform Gaussian projections in reducing tensor dimensions.
problem Reducing the dimension of high-dimensional tensors for machine learning.
method Tensorized Rademacher random projections using Tensor Train decomposition.
result Tensorized Rademacher projections can replace Gaussian projections in tensor compression.
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 man…
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 dim…
As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study th…
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 …
FROCC uses random projections for fast one-class classification.
problem Efficient one-class classification for large datasets.
method Random projection of data onto random unit vectors, with region bounding.
result Significant improvement in ROC performance with substantial speedup.
A new method approximates the Sliced-Wasserstein distance without random projections.
problem Efficiently approximating the Sliced-Wasserstein distance for machine learning applications.
method Utilizing the concentration of measure phenomenon to develop a deterministic approximation.
result The approximation error goes to zero as the dimension increases, under a weak dependence condition.
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 …
Projected random forests improve circular data prediction with adaptive arc length and finite-sample coverage.
problem Regression with circular responses.
method Adapting linear-response models to circular data using projection and random forest out-of-bag mechanism.
result Projected random forest out-of-bag conformal prediction sets are more efficient and shorter than alternative methods.
Random projections improve classifier generalization without needing to choose the best threshold.
problem Improving classifier generalization without choosing the best threshold.
method Thresholding a random one-dimensional feature after random projection of data.
result Generalization gap is significantly smaller than linear classifiers.
Unified theory and debiasing framework for random oblique projections in high dimensions.
problem Systematic statistical bias in random oblique projections induced by sampling.
method Unified non-asymptotic theory and debiasing framework.
result Sharp bias--variance characterizations and improved approximation accuracy.
Improved neural network training in low-dimensional random bases.
problem Inefficient optimization in large-scale neural networks.
method Re-draw random subspace at each training step, apply independent projections to different network parts.
result Significantly better optimization performance and efficiency.
Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-k projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These condit…
Modified neural network enhances unsupervised anomaly detection.
problem Unsupervised anomaly detection in multimodal data.
method Neural network with modified random projection outlyingness.
result Performance comparable to state-of-the-art methods.
Performing signal processing tasks on compressive measurements of data has received great attention in recent years. In this paper, we extend previous work on compressive dictionary learning by showing that more general random projections may be used, including sparse ones. More precisely, we examine compressive K-mean…
We adapt the idea of random projections applied to the output space, so as to enhance tree-based ensemble methods in the context of multi-label classification. We show how learning time complexity can be reduced without affecting computational complexity and accuracy of predictions. We also show that random output spac…
Sharp-SSL uses random projections to identify important variables for semi-supervised learning.
problem High-dimensional semi-supervised learning problems.
method Careful aggregation of low-dimensional results from many axis-aligned random projections.
result Sharp-SSL algorithm can recover signal coordinates with high probability.
New algorithm uses random projections for robust, sparse data classification.
problem Improving robustness and sparsity in data classification.
method Randomly projects data into a high-dimensional space, truncates small entries, and applies a cap operation.
result The method enhances classification accuracy with minimal loss, especially in noisy conditions.
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 th…
We study the use of "sign α-stable random projections" (where 0<α≤2) for building basic data processing tools in the context of large-scale machine learning applications (e.g., classification, regression, clustering, and near-neighbor search). After the processing by sign stable random projections, the inner pr…
Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic and real data.
Subsampled Randomized Hadamard Transform (SRHT), a popular random projection method that can efficiently project a d-dimensional data into r-dimensional space (r≪d) in O(dlog(d)) time, has been widely used to address the challenge of high-dimensionality in machine learning. SRHT works by rotating the input …
Paper improves performance guarantees for Rademacher projections.
problem Improving statistical guarantees for Rademacher random projections.
method Algebraic framework for proving Schur-concavity properties.
result Novel Schur-concavity property of Rademacher projections with improved performance.
We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
New framework reduces private mean estimation error with optimal efficiency.
problem Locally private mean estimation of high-dimensional vectors.
method ProjUnit framework: random projections, normalization, and optimal algorithm execution in lower dimensions.
result Optimal error up to a 1+o(1)-factor with computational efficiency and low communication complexity.
A method for clustering small datasets in high dimensions using random projections.
problem Challenges in clustering small datasets in high-dimensional spaces.
method Random projection followed by binary clustering in one-dimensional space.
result Statistically significant clustering structures can be found with as few as 100-200 points.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
TTRP method preserves distances in high-dimensional data with reduced storage and speed.
problem Preserving distances in high-dimensional datasets efficiently and accurately.
method Tensor train random projection (TTRP) using TT-ranks of one.
result TTRP is an expected isometric projection with bounded variance.
Develops precise expressions for random projections for better machine learning tasks.
problem Improving the accuracy of dimensionality reduction in machine learning tasks.
method Exploits recent developments in spectral analysis of random matrices to derive accurate expressions for random projection matrices.
result Provides precise expressions that reflect the practical performance of sketching methods, including Gaussian and Rademacher sketches.
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…