A new pruning method reduces neural network computation without retraining.
problem Efficiently reduce neural network computation while maintaining accuracy.
method Structured directional pruning via perturbation orthogonal projection.
result Achieves state-of-the-art pruned accuracy without retraining.
Characterizes the width of real projective spaces and computes Morse index.
problem Finding the minimum area of hypersurfaces in real projective spaces.
method Uses min-max width and Morse index calculations on Clifford hypersurfaces.
result Characterizes the first min-max width of real projective spaces.
Projective preferential Bayesian optimization learns user preferences in high dimensions.
problem Finding extrema of a black-box function in high-dimensional spaces.
method Projective preferential queries for feedback in human-interaction.
result Framework finds global minimum of high-dimensional black-box function.
In this paper, we study the Yang-Mills functional on quantum Heisenberg manifolds using the appratuses developed by A. Connes and M. Rieffel. It is discovered that a connection on a projective module over a quantum Heisenberg manifold is a minimum of Yang-Mills functional whicih is a critical point that is different wi…
Inflating the minimum norm interpolator improves linear regression generalization error.
problem Highly anisotropic covariances and diverging d/n in linear regression. method Inflating the minimum ℓ2 norm interpolator by a constant greater than one. result Inflating the minimum norm interpolator improves generalization error.
We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other …
This paper identifies knot projections with reductivity two.
problem Determining knot projections with a specific reductivity level.
method Examined four types of reductivity (Seifert type splice, non-Seifert type splice, recursively, simultaneously) and their combinations.
result Identified all knot projections with reductivity two for the four definitions.
This work improves understanding of projection robust optimal transport distances.
problem Understanding the behavior of minimum Wasserstein estimators in high-dimensional and misspecified models.
method Adopting projection robust (PR) optimal transport, establishing statistical properties, proposing IPRW distance, and providing asymptotic guarantees.
result Established fundamental statistical properties and proposed new distances that outperform Wasserstein distances empirically.
The paper derives formulas for braid index of alternating links.
problem Determining the braid index of alternating links.
method Diagrammatic approach based on minimum projections.
result Explicit formulas for braid index of many alternating links.
Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot K i…
Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classificati…
If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.
Project predicts stock performance and builds an efficient portfolio for six Indian sectors.
problem Predicting stock prices accurately for optimal portfolio design.
method Analysis of time series, machine learning, and deep learning models; Modern Portfolio Theory; minimum variance and optimal risk portfolio optimization.
result Built and tested an efficient portfolio for six Indian sectors using historical stock prices.
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.
Develops efficient nonparametric testing with random projections.
problem High computational complexity in nonparametric inference with large data.
method Random projection strategy for kernel ridge regression.
result Achieves testing optimality with minimum number of projections.
Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot…
The paper characterizes new invariant spinr spinors on projective spaces.
problem Characterizing new invariant spinr spinors on projective spaces. method Adapting spin representation via exterior forms to the generalised spinr context. result Complete description of the space of invariant spinr spinors for CPn, HPn, and OP2. Batching stabilizes risk in high-dimensional linear regression models.
problem Stability and risk behavior in high-dimensional overparameterized linear regression.
method Minimum-norm overparameterized linear regression model with batch-partitioning.
result Optimal batch size is inversely proportional to noise level and overparametrization ratio, leading to stable risk behavior.
New method estimates minimizer and minimum value of a regression function.
problem Estimating minimizer and minimum value of a regression function from noisy data.
method Projected gradient descent with gradient estimated by regularized local polynomial algorithm, followed by a rate optimal nonparametric procedure.
result Achieves minimax optimal rates of convergence for smooth and strongly convex functions.
Characterizes submanifolds with minimum ratio of diameter to focal radius.
problem Finding submanifolds with the minimum ratio of extrinsic diameter to focal radius.
method Combining K. Sakamoto's classification of submanifolds with planar geodesics and A. Schur's Bow Lemma for space curves.
result Essentially round spheres or Veronese embeddings of projective spaces achieve the minimum ratio.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…
The paper finds petal numbers of torus knots using superbridge indices.
problem Determining petal numbers of torus knots.
method Using superbridge indices, the paper establishes relations between superbridge indices and petal numbers of torus knots.
result The petal number of Tr,s is found to be 2s−1 when 1<r<s and r≡1mods−r. The upper bound is $2s - 2\Big\lfloor \frac{s}{r} \Big
floor +1$. We study alignment in linear neural networks and its relation to gradient descent.
problem Understanding alignment in linear neural networks and its impact on training.
method Defined alignment for fully connected networks, analyzed alignment under gradient descent, and compared gradient descent to projected gradient descent for layer-constrained networks.
result Gradient descent can converge linearly to a global minimum when alignment is invariant, and alignment is impossible with large datasets in layer-constrained networks.
Weight normalization and reparametrized gradient descent adaptively regularize weights and converge to minimum l2 norm solutions.
problem Adapting to non-convex weight normalization for convergence to minimum l2 norm solutions.
method Weight normalization and reparametrized projected gradient descent (rPGD) for overparametrized least-squares regression.
result rPGD converges close to the minimum l2 norm solution, even for far-from-zero initializations.
Project predicts stock prices for robust portfolio design in Indian sectors.
problem Precise stock price prediction for robust portfolio design.
method Minimum variance and optimal risk portfolio optimization using past stock prices.
result Backtesting shows improved performance of optimized portfolios over equal weight portfolio.
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
This paper defines RII number for knot projections and shows it can be any nonnegative number.
problem Defining and quantifying the minimum number of specific types of deformations for knot projections.
method Using deformations of types 1, 2, and 3, analogs of Reidemeister moves, to simplify knot projections and define RII number.
result RII number can be any nonnegative number, not just zero as previously conjectured.
We present algorithms for topic modeling based on the geometry of cross-document word-frequency patterns. This perspective gains significance under the so called separability condition. This is a condition on existence of novel-words that are unique to each topic. We present a suite of highly efficient algorithms based…
We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, whic…
We propose a projected gradient dynamical system as a model for a bargaining scheme for an asset for which the two interested agents have personal valuations which do not initially coincide. The personal valuations are formed using subjective beliefs concerning the future states of the world and the reservation prices …
The paper encodes local shapes of polynomial curves using permutations.
problem Measuring non-convexity of real algebraic plane curves.
method Generic projections avoiding specific tangencies.
result Local shapes of curves can be encoded in alternating permutations.
Minimal area of Teichmüller curves in genus two is found to be 3π/5.
problem Finding the minimum hyperbolic area of Teichmüller curves in genus two.
method Combining small-area classification of orbifolds and affine descent construction for quadratic differentials, excluding patterns by arithmetic, marked-point, and covering obstructions.
result The minimum hyperbolic area of Teichmüller curves in genus two is 3π/5.
New method improves MMD estimation without convexity assumptions.
problem Lack of theoretical guarantees for MMD estimation algorithms.
method Preconditioned gradient descent (PGD) scheme for MMD optimization.
result PGD scheme converges globally under specific conditions.
In this paper we continue to study (`strong') Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with c…
With increasing concerns about security, the need for highly secure physical biometrics-based authentication systems utilizing \emph{cancelable biometric} technologies is on the rise. Because the problem of cancelable template generation deals with the trade-off between template security and matching performance, many …
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
Method selects interpretable circular coordinates from data.
problem Abstract circular coordinates are hard to interpret.
method Minimum-weight basis problem in vector matroid for selecting interpretable circle-valued coordinates.
result Proves consistency of cochain inner product estimator.
The study analyzes how covariance estimation errors affect the global minimum-variance portfolio under heavy-tailed distributions.
problem The impact of covariance estimation errors on the global minimum-variance portfolio under heavy-tailed distributions.
method Characterization of covariance-estimation error's effect on GMVP suboptimality, derivation of regret identity and bound, application to heavy-tailed returns.
result The decision geometry of GMVP regret is invariant to a (p-1)-dimensional projection of the error matrix, with invariance to the covariance-scale direction as an exact special case.
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
New method finds optimal learning rates for neural nets.
problem Finding optimal learning rates in stochastic neural networks.
method Gradient-only line searches using Non-negative Associative Gradient Projection Points (NN-GPPs).
result Learning rates can be reliably resolved as step sizes along search directions.
Research examines GMIB and reset options in variable annuities.
problem Understanding the value and rationality of GMIB and reset options.
method Exploration of various parameters affecting GMIB value and calculation of critical future interest rates for reset option rationality.
result Insight into how future market performance and interest rates influence policyholder and insurer actions.
Three LF training criteria improve neural network acoustic models without cross-entropy pre-training.
problem Improving purely sequence-trained neural network acoustic models.
method Comparison of three lattice-free discriminative training criteria (MMI, bMMI, sMBR) on LVCSR tasks.
result LF-bMMI models outperform plain LF-MMI models by 5% WER on Switchboard datasets.
SMAVE optimizes SDR by projecting onto a low-dimensional subspace on a Riemannian manifold.
problem High-dimensional regression challenges due to the curse of dimensionality.
method SMAVE combines nearest-neighbor localization and Riemannian stochastic gradient ascent.
result SMAVE achieves almost-sure convergence and matches RMAVE's synthetic subspace recovery rate.
New method clusters data points by finding optimal directions.
problem Subspace clustering problem, especially in noisy and close subspaces.
method Optimal direction search via convex program, alternating direction method of multipliers.
result Significantly outperforms existing methods, especially in noisy scenarios.
Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal Z22-equivariant triangulations of 2-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{R…
The paper studies energy functionals for Lagrangian tori in complex projective space.
problem Investigating energy functionals for Lagrangian tori in complex projective space.
method Introducing an energy functional based on the potential of associated Schrödinger operators and studying its behavior on specific families of tori.
result Proposes that the minimum of the energy functional is achieved by the Clifford torus.
A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…
We describe a new technique for computing lower-bounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are represented in terms of …