Two-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to biva…
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169,051 papers · 148 categories
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48 results for “Linear projection”
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.
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.
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.
The paper tackles linear bandits with projections, achieving optimal regret.
problem Learning the best arm in a linear bandit model with unobservable projection rewards.
method Developed strategies for both finite and infinite arms, achieving optimal regret bounds.
result Achieved optimal regret bounds for both finite and infinite arms.
New insights into continual learning for deep models, showing convergence issues but local linear solutions.
problem Challenges in continual learning for homogeneous deep models.
method Sequential projections onto task margin sets, leveraging nonconvex projection theory.
result Local linear convergence under certain conditions for homogeneous deep networks.
We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the …
Linear and Regular Kepler-Manev Dynamics via Projective Transformations: A Geometric Perspectivemath-ph
Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.
problem Regularizing and linearizing nonconservative central force dynamics.
method Projective transformation and conformal scaling in configuration and phase spaces.
result Full linearization of Kepler and Manev dynamics in any finite dimension.
A method to interpret neural networks by isolating active subnetworks.
problem Interpreting the complex computations of deep neural networks.
method Switched linear projections to decompose network activity.
result Patterns in deactivated neurons are meaningful for network understanding.
Paper studies PSGD for constrained optimization problems and its statistical properties.
problem Online inference for constrained optimization problems.
method Stochastic gradient descent with projection (PSGD) for constrained optimization.
result Limiting distribution of PSGD-based estimates under linear-equality constraints.
Computes -algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes -algebroid structure.
result Provides invariants and constant-rank replacements of singular foliations.
The paper suggests using -separable projections instead of linear separability for machine learning.
problem Complex data with multimodal distributions are difficult to learn using linear separability.
method Use -separable projections to simplify the learning process.
result Projection on line segments makes the learning process easier for complex data.
New conditions ensure points can be uniquely represented by combinations of variety elements.
problem Ensuring points can be uniquely represented by combinations of variety elements.
method Conditions on contact locus of general linear spaces.
result Conditions ensuring non tangential weak defectiveness of projective varieties.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
The paper classifies and decomposes quaternionic projective transformations.
problem Classifying and decomposing elements of the projective linear group .
method Algebraic characterization of dynamical types using reversibility, decomposition of elements into simple elements.
result Offered a complete classification for elements of .
We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Bas…
The paper analyzes an ensemble of randomly projected linear discriminants for high-dimensional data.
problem Classification issues in small samples of high-dimensional data.
method Asymptotic analysis using random matrix theory.
result The ensemble offers a performance advantage under certain conditions.
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.
H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…
The paper proves a flat torus theorem for certain groups acting on convex domains.
problem Establishing a flat torus theorem for specific groups acting on convex domains.
method Analyzing discrete groups in acting convex co-compactly on a properly convex domain.
result An analogue of the flat torus theorem for spaces is proven for these groups.
For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.
Study calculates Kulkarni limit sets for quaternionic projective groups.
problem Computing Kulkarni limit sets for quaternionic projective groups.
method Natural action of quaternionic projective linear group on quaternionic projective space.
result Computed Kulkarni limit sets for cyclic subgroups.
Efficiently projects points onto polytopes, especially useful in web-scale applications.
problem Efficiently projecting points onto polytopes in large-scale applications.
method Developed a vertex-oriented incremental algorithm for polytope projection, tailored for simplex and unit-box cut polytopes.
result Majority of projections lie on vertices of polytopes, leading to significant performance improvements.
Improved regret bounds for scalable bandit convex optimization.
problem Designing online algorithms for high-dimensional bandit convex optimization.
method Projection-free algorithms using a linear optimization oracle.
result First algorithm with expected regret in calls.
New algorithm improves convergence rates for convex optimization problems.
problem Convex optimization problems with noisy stochastic data.
method Stochastic proximal point algorithm with weak linear regularity condition.
result Achieves $\mathcal{O}\left(\frac{1}{k}
ight)$ convergence rate for SPP.
End-to-end CCA optimizes both discriminative and latent space projections for multi-view learning.
problem Lack of class label information in CCA for multi-view learning tasks.
method Simultaneously optimizes a CCA-based and a task objective in an end-to-end manner to learn a non-linear CCA projection.
result Significant improvement in cross-view classification, regularization with a second view, and semi-supervised learning.
Here, a non-linear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.
PPF uses projections to improve classification accuracy.
problem Improving classification accuracy in multi-class problems.
method PPF constructs trees using projections of variables, enhancing traditional random forest.
result PPF outperforms traditional random forest in multi-class problems.
Improved SRHT for linear SVM classification with higher accuracy.
problem Inefficient random projection methods for high-dimensional data.
method Importance sampling and deterministic top- sampling for effective low-dimensional embedding.
result Higher classification accuracy on real-life datasets.
New method converts LVAs into linear projections for better understanding of complex models.
problem Limited interpretability of nonlinear machine learning models.
method Animated linear projections and radial tours.
result Improved understanding of variable importance in complex models.
LightOn OPUs accelerate randomized numerical linear algebra, reducing computational costs.
problem Computational bottleneck in randomization step for large-scale linear algebra.
method Near constant-time linear random projections from LightOn OPUs.
result Significant acceleration of RandNLA algorithms with negligible precision loss.
MILDA uses unlabelled data to compute LDA projections.
problem Training LDA models with unlabelled data.
method Minimal prior information to compute LDA projection vector.
result MILDA closely matches supervised LDA performance and adapts to non-stationary data.
Study of magnetic curves in SL(2,R) with quantization and horocycle projections.
problem Investigating magnetic curves in the real special linear group SL(2,R).
method Geodesics of Hopf tubes, quantization of periodic curves, and projection to horocycles.
result Quantization of periodic contact magnetic curves in SL(2,R) to rational numbers.
The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions …
The study provides a criterion for fractional-linear integrals of geodesics on surfaces.
problem Existence and classification of fractional-linear integrals for geodesic flows on Riemannian surfaces.
method Criterion and analysis of moduli space of local integrals.
result The moduli space of such local integrals is either the 2D projective plane or finite points.
A new method optimizes projection directions for sliced Wasserstein distances.
problem Finding informative projecting directions for sliced Wasserstein distances is computationally expensive.
method Amortized projection optimization to predict directions efficiently.
result Proposed amortized models improve generative modeling performance.
New algorithm reduces online learning iterations by a factor of T^2/3.
problem Efficiency in online learning with smooth cost functions.
method Follow-the-Perturbed-Leader method using online primal-dual framework.
result Guaranteed T^2/3 regret for general online convex optimization.
Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank- 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…
Generic linear perturbationsmath.GT
In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear pert…
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…
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.
Paper improves multi-objective optimization using machine learning and KSA.
problem Improving performance of multi-objective optimization solutions.
method Employed machine learning to identify the best projected space for KSA.
result Up to 12% improvement in time achieved through learning method.
Liouville classification of integrable geodesic flows on a projective plane in a potential fieldmath.DG
Classifies geodesic flows on projective plane with potential field.
problem Classifying geodesic flows on a projective plane with a potential field.
method Liouville classification and calculation of Fomenko--Zieschang invariants.
result All Fomenko--Zieschang invariants of the system are calculated.
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and on…
Paper tackles online learning on curved spaces without projections.
problem Online learning on Riemannian manifolds with computational constraints.
method Develops projection-free algorithms for geodesically convex optimization.
result Achieves sub-linear regret guarantees in online geodesically convex optimization.
This work analyzes privacy-utility trade-offs in linear regression with noise and projections.
problem Balancing privacy and utility in machine learning models trained on private data.
method Analyzes two schemes: additive noise and random projections, using differential privacy based on conditional mutual information.
result Projecting data to a lower-dimensional subspace before adding noise yields a better privacy-utility trade-off.
In this paper, we propose and study random maxout features, which are constructed by first projecting the input data onto sets of randomly generated vectors with Gaussian elements, and then outputing the maximum projection value for each set. We show that the resulting random feature map, when used in conjunction with …
We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction …