Paper extends RPD for better handling multiple modalities and non-convexity.
problem Handling multiple modalities and non-convexity in data clouds.
method Computes RPD in a reproducing kernel Hilbert space using kernel principal component analysis.
result The method outperforms RPD and is comparable to other models on benchmark datasets.
The paper studies randomized approximations of Tukey's depth for log-concave isotropic data.
problem The challenge of approximating Tukey's depth in high dimensions.
method The study examines randomized algorithms for approximating Tukey's depth for log-concave isotropic data.
result Randomized algorithms correctly approximate maximal depth and close to zero depths but not intermediate depths.
Investigates projections onto explicit subspaces and their variance effects.
problem Understanding the variance preservation in explicit subspace projections.
method Investigates projections onto explicit subspaces of varying dimensionality and analyzes the variance effects.
result Developed new bounds for Euclidean distances and inner products.
New algorithm learns random neural networks efficiently.
problem Learning random constant-depth neural networks efficiently.
method Presented a PTAS (Polynomial-Time Approximation Scheme) for learning random Xavier networks of fixed depth.
result For any fixed ε and depth i, there is a poly-time algorithm that learns random Xavier networks up to an additive error of ε.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
Introduces Polar Depth for analyzing multivariate heavy-tailed data extremes.
problem Analyzing the behavior of extremes from multivariate heavy-tailed distributions.
method Introduces Polar Depth, a novel statistical depth function expressed in polar coordinates.
result The polar depth of the largest observations converges to the polar depth of the limiting distribution as the threshold increases.
New methods improve prediction performance and reduce computation time in boosting and random forest models.
problem Improving prediction performance and reducing computation time in boosting and random forest models.
method Random tree depth injection approach for Boosting and Random Forests.
result The new methods can improve prediction performance and reduce computation time by up to 40%.
Deep forests enhance expressiveness exponentially with depth, not width or tree size.
problem Understanding the role of depth, width, and tree size in deep forest performance.
method Provided upper and lower bounds on deep forest approximation complexity.
result Depth exponentially enhances deep forest expressiveness.
Study on neural network initialization with shaped infinite depth-and-width networks.
problem Understanding the distribution of random covariance matrices in shaped infinite-depth-and-width networks.
method Introduced the Neural Covariance SDE to model the distribution of the random covariance matrix.
result Identified the precise scaling of the activation function necessary for a non-trivial limit.
We study the behavior of untrained neural networks whose weights and biases are randomly distributed using mean field theory. We show the existence of depth scales that naturally limit the maximum depth of signal propagation through these random networks. Our main practical result is to show that random networks may be…
In this paper we study the learnability of deep random networks from both theoretical and practical points of view. On the theoretical front, we show that the learnability of random deep networks with sign activation drops exponentially with its depth. On the practical front, we find that the learnability drops sharply…
The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a comp…
Dropout and RaM become equivalent in large ResNets as depth and width increase.
problem Improving performance in deep learning models.
method Comparing Dropout and Random Gradient Masking in ResNets.
result Dropout and RaM converge to the same large-scale limiting dynamics in ResNets.
Self-attention models benefit equally from width and depth, but beyond a certain point, depth becomes less efficient.
problem Understanding the optimal balance between depth and width in self-attention models.
method Theoretical predictions and empirical ablations on networks of varying depths and widths.
result An optimal width of 30K is recommended for a 1-Trillion parameter network, marking a significant width for self-attention models.
Paper uses statistical depth to create DP estimators for regression.
problem Creating differentially private estimators in high dimensions.
method Uses halfspace and regression depth to analyze maximum influence and construct DP estimators.
result New DP estimators for location and regression show favorable performance.
The study analyzes deep linear networks from random initialization, capturing dynamics and hyperparameter effects.
problem Understanding training dynamics in deep linear networks from random initialization.
method Theoretical analysis of gradient descent dynamics in deep linear networks with random initialization and large data.
result Captures the 'wider is better' effect and hyperparameter transfer effects, contrasting with neural-tangent parameterization.
We derive finite width and depth corrections for the Neural Tangent Kernel (NTK) of ResNets and DenseNets. Our analysis reveals that finite size residual architectures are initialized much closer to the "kernel regime" than their vanilla counterparts: while in networks that do not use skip connections, convergence to t…
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.
Approximates large Random Forest models to save space.
problem Large model size limits deployment on small devices.
method Approximate each tree using multinomial logistic regression.
result Reduced model size without significant loss of accuracy.
Adversarial examples in deep ReLU networks with constant depth.
problem Adversarial perturbations lead to large output changes in deep ReLU networks.
method Analyzes the phenomenon in networks with independent Gaussian parameters and constant depth.
result Adversarial examples arise due to functions being close to linear.
Improved isolation forest for better outlier detection.
problem Outlier detection in multivariate data.
method Random cuts across feature space, size of feature space, and point assignment information.
result Improved results in many situations without modifying tree structure.
Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.
problem Approximating functions with depth 2 networks in high dimensions.
method Lower bound proof using worst-to-average-case random self-reducibility.
result Proves depth 2 networks can't approximate certain functions as well as depth 3 networks, resolving an open problem.
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.
Enhanced ODT with Feature Concatenation boosts learning efficiency.
problem Insufficient learning efficiency of ODT due to linear projections not being transmitted to child nodes.
method Feature Concatenation ( exttt{FC-ODT}) to transmit linear projections along decision paths.
result Experiments show exttt{FC-ODT} outperforms state-of-the-art decision trees with a limited tree depth.
seMCD computes depth functions with statistical guarantees using sequential Monte Carlo.
problem Computing depth functions is computationally challenging, especially in high dimensions.
method Sequential Monte Carlo methodology with theoretical and empirical guarantees.
result The seMCD method provides accurate depth approximations with fewer samples than traditional methods.
New algorithm quantifies uncertainty in regression models for complex data types.
problem Uncertainty quantification in regression models for complex data types.
method Model-free uncertainty quantification algorithm based on conditional depth measures and kernel mean embeddings.
result Provides faster convergence rates and non-asymptotic guarantees for prediction regions.
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.
This work improves the lottery ticket hypothesis by reducing over-parameterization requirement.
problem Approximating a neural network by pruning a randomly over-parameterized network.
method Connecting pruning ReLU networks to extsc{SubsetSum} problem, showing logarithmic over-parameterization sufficiency.
result Logarithmic over-parameterization is sufficient for approximating any target neural network.
Study reveals how depth of reasoning affects generalization in models.
problem Understanding scaling behavior of generalization with CoT depth.
method Theoretical model of CoT in linear regression using random matrix theory.
result Sharp phase transition between exponential and polynomial improvement, saturation, and overthinking.
ReLU networks don't exponentially distort curve lengths as previously thought.
problem Understanding how neural networks distort curve lengths with depth.
method Analyzing expected length distortion of ReLU networks with random initialization.
result Expected length distortion does not grow with depth, and shrinks slightly.
We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one…
We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features…
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…
A new depth function improves multivariate data analysis by considering variability directions.
problem Developing a depth function that respects quantile properties and is affine-invariant.
method Integrating rank-weighted depth with affine-invariance and covariance matrices.
result The AI-IRW depth function provides accurate quantile estimates and is robust to data variability.
Randomly initialized ReLU networks of depth two can approximate smooth functions well.
problem Approximation power of two-layer networks of random ReLUs.
method Harmonic analysis and ridgelet representation theory for upper bounds, dimensionality arguments for lower bounds.
result Near-matching upper and lower bounds for L2-approximation and Sobolev norms. 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.
Batch normalization makes deep neural networks' representations increasingly orthogonal.
problem Orthogonality of deep neural network representations.
method Random linear transformations in successive batch-normalizations.
result Orthogonality of representations improves SGD performance.
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.
Scene recognition with RGB images has been extensively studied and has reached very remarkable recognition levels, thanks to convolutional neural networks (CNN) and large scene datasets. In contrast, current RGB-D scene data is much more limited, so often leverages RGB large datasets, by transferring pretrained RGB CNN…
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.
In the last decade, supervised deep learning approaches have been extensively employed in visual odometry (VO) applications, which is not feasible in environments where labelled data is not abundant. On the other hand, unsupervised deep learning approaches for localization and mapping in unknown environments from unlab…
Study on learning rates in neural networks of varying depth.
problem Dependence of maximal update learning rate on network depth.
method Analysis of random fully connected ReLU networks with mean-field weight initialization.
result Maximal update learning rate scales like L−3/2 with network depth. Study on critical points in random neural networks, revealing three regimes based on activation function.
problem Investigating the expected number of critical points in random neural networks.
method Deriving asymptotic formulas for critical points under infinite-width limit and suitable regularity conditions.
result Three distinct regimes of critical points behavior depending on activation function.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
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.
New ODE-Block handles stateful layers with continuous-in-depth functions using basis functions.
problem Handling stateful layers in ODE-Nets.
method Formulate ODE-Block using continuous-in-depth functions with basis function expansions.
result Enables state-of-the-art performance and reduces memory footprint.
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.