Study on detecting a single spike in high-dimensional data matrices.
problem Detecting a single unknown spike in high-dimensional rectangular data matrices.
method Analysis of likelihood ratio between spiked and null models, using Gaussian fluctuations and Talagrand's interpretation of cavity method.
result Asymptotic Gaussian fluctuations of the likelihood ratio below the BBP threshold, with open maximal parameter region.
Improves signal detection in non-Gaussian noise using transformed data.
problem Signal detection in rank-one signal-plus-noise data matrices.
method Pre-transforming matrix entries and using linear spectral statistics for hypothesis testing.
result Sharp phase transition of largest eigenvalues in spiked rectangular matrices.
New algorithm for signal estimation in noisy matrix models.
problem Signal estimation in rectangular spiked matrix models with rotationally invariant noise.
method Orthogonal Approximate Message Passing (OAMP) algorithm for signal estimation.
result Optimal OAMP algorithm minimizes mean-squared error and achieves Bayes-optimal performance.
PLS-SVD struggles with missing data in multimodal datasets, showing a phase transition in performance.
problem Missing data in PLS-SVD for multimodal datasets.
method Replica-symmetric analysis of spiked rectangular random matrices with missing entries.
result PLS-SVD performance transitions from uninformative to informative singular vectors at a critical signal-to-noise threshold.
Paper studies S-rectangular DR-RL models for robust reinforcement learning with near-optimal sample complexity.
problem Addressing distributional discrepancies in reinforcement learning environments.
method Empirical value iteration algorithm for divergence-based S-rectangular DR-RL models.
result Near-optimal sample complexity bound of O(∣S∣∣A∣(1−γ)−4ε−2). Rectangular diagrams help analyze foliations in 3-sphere.
problem Analyzing foliations in 3-sphere complements.
method Introduced rectangular diagrams for foliations and links.
result Any co-orientable finite depth foliation can be presented by a compatible rectangular diagram.
Rectangular Bounding Process (RBP) improves partitioning efficiency in multi-dimensional spaces.
problem Creating many unnecessary divisions in sparse regions when describing dense regions.
method Introduces Rectangular Bounding Process (RBP) to efficiently partition multi-dimensional spaces using a bounding strategy.
result The RBP is self-consistent and can be extended to infinite space, offering rich yet parsimonious expressiveness.
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Researchers extend knot theory formulas to non-rectangular cases.
problem Applying universal-matrix precursor formulas to non-rectangular knot representations.
method Reformulated previously known formulas for simplest non-rectangular representations [r,1].
result Demonstrated drastic simplification of formulas after reformulation.
Improved bounds for knot crossings in different mosaic patterns.
problem Finding tighter bounds for knot crossings in rectangular and hexagonal mosaics.
method Extended Howard and Kobin's proof to hexagonal mosaics and shortened the rectangular proof.
result New bounds for hexagonal mosaics with improved efficiency in rectangular mosaics.
New proof for symmetric spaces with rectangular lattices.
problem Characterizing symmetric spaces with rectangular unit lattices.
method Explicit construction of isometric embeddings and analysis of root systems.
result Symmetric spaces with rectangular unit lattices are symmetric R-spaces.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
Proves a theorem for comparing surfaces in 3D space.
problem Comparing isotopy classes of compact surfaces in 3-sphere.
method Uses rectangular diagrams to formalize and compare surfaces.
result Proves a Reidemeister type theorem for rectangular diagrams of surfaces.
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need…
In this paper Legendrian graphs in (R3,ξst) are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendr…
The study improves inequalities for link diagrams and introduces weak rectangular diagrams.
problem Improving inequalities for link diagrams and understanding their properties.
method Introducing weak rectangular diagrams and proving new inequalities.
result Generalizes and subsumes many known inequalities related to multi-crossing numbers.
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on S3 and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in p…
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
Study reveals 1/f noise in signals made from nonoverlapping rectangular pulses.
problem Analyzing 1/f noise in signals composed of nonoverlapping pulses. method Derived a general formula for power spectral density, analyzed rectangular pulse case.
result Observed pure 1/f noise until very low frequencies with long pulse durations. Paper studies robust MDPs, improving sample complexity and asymptotic performance.
problem Optimal robust policy and value function in robust MDPs with generative models.
method Improves prior results on non-asymptotic and asymptotic performances of robust MDPs, considering various uncertainty sets.
result Improved sample complexity and asymptotic normality of optimal robust value function.
New formula simplifies evolution of twist knots and calculates Racah matrices for rectangular representations.
problem Simplifying evolution of twist knots and calculating Racah matrices for rectangular representations.
method Developed a universal formula for triangular evolution matrix B applicable to rectangular representations R=[rs]. Used skew characters and Macdonald polynomials. result Explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix Sˉ in arbitrary rectangular representation R. If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I. A. Dynnikov. Using this, Henrich and Kauffman gave an up…
A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show …
Optimal unimodal fitting for linear loss functions in a sequential, efficient manner.
problem Optimal unimodal transformation of univariate model scores under linear loss functions.
method Proposes a sequential approach to estimate the optimal rectangular fit for observed samples with each new sample.
result Sequential approach achieves optimal efficiency with logarithmic time complexity per iteration.
Single-spike neurons can approximate as well as multi-spike neurons.
problem Limitation of single-spike neurons in spiking neural networks.
method Comparison of single-spike and multi-spike neural networks.
result Single-spike and multi-spike neural networks are equivalent in approximation capabilities.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. LoRA fine-tuning creates intruder dimensions that can cause forgetting, and a new law predicts when this happens.
problem Predicting when LoRA fine-tuning creates intruder dimensions that can cause catastrophic forgetting.
method Derived a per-layer critical update strength s∗ and an exact secular-equation characterization of the updated spectrum. result The law localizes the empirical threshold within a factor of two on 82% of layers and separates intruder-bearing from intruder-free layers at deployment.
Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.
problem Uniqueness of 3D shape of rectangular Clifford torus based on isoperimetric ratio.
method Closed-form formulas for isoperimetric ratio of stereographic projection, strict monotonicity.
result Isoperimetric ratio does not uniquely determine rectangular Clifford torus shape.
New transformations preserve link isotopy, showing complexity differences.
problem Link isotopy preservation with complexity differences.
method Introducing multiflypes of rectangular diagrams of links.
result Two diagrams of same complexity not related by simpler moves.
The paper finds new constrained Willmore minimizers for non-rectangular tori.
problem Finding constrained Willmore minimizers for non-rectangular tori.
method Analyzing immersed tori in 3-space to minimize Willmore energy.
result The candidates constructed in previous work are constrained Willmore minimizers in certain non-rectangular conformal classes.
Improved gradient descent for rectangular matrix completion without ℓ2,∞ regularization.
problem Nonconvex rectangular matrix completion without ℓ2,∞ regularization. method Gradient Descent without ℓ2,∞ regularization. result Improved sampling rate from O(poly(κ)μ3r3log3n/n) to O(μ2r2κ14logn/n). Much of studies on neural computation are based on network models of static neurons that produce analog output, despite the fact that information processing in the brain is predominantly carried out by dynamic neurons that produce discrete pulses called spikes. Research in spike-based computation has been impeded by th…
Paper trains SNNs for classification using first-to-spike decoding.
problem Training SNNs for classification under GLM model.
method Proposes first-to-spike decoding method for SNNs.
result Improves accuracy and efficiency of SNN classification.
The paper examines how spike strengths and alignments affect overfitting in linear regression models.
problem The impact of spike strengths and alignments on overfitting in linear regression models.
method Characterization of generalization error through exact expressions and analysis of spike strengths, aspect ratio, and target alignment.
result Increasing spike strength can lead to catastrophic overfitting before benign overfitting, especially in well-specified aligned problems.
New method improves neural spike train models by minimizing divergence directly, leading to better performance.
problem Poor performance and divergence issues in spike train models using maximum likelihood estimation.
method Directly minimize maximum mean discrepancy using spike train kernels and stochastic optimization.
result The proposed method generates well-behaved models with better control over feature trade-offs.
Study inequalities for singular values of rectangular matrices.
problem Inequalities for singular values of rectangular matrices.
method Study convex cones associated to isotropic representations of symmetric spaces.
result Describe inequalities by cohomological conditions.
SNNs enhance high-frequency price spike forecasting in HFT environments.
problem Conventional financial models fail to capture fine temporal structure in high-frequency price spikes.
method Application of Spiking Neural Networks (SNNs) with hyperparameter tuning via Bayesian Optimization (BO).
result SNN models optimized with PSA achieve significantly higher cumulative returns in backtesting.
Deep learning classifies knots using rectangular diagrams.
problem Recognizing and distinguishing knots, especially the unknot.
method Represent knots as rectangular Dynnikov diagrams and use neural networks to classify them.
result Neural networks can effectively distinguish knots from each other.
Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its…
Bayesian method detects neuron spike activities from noisy fluorescence data.
problem Detecting neuron spike activities from noisy fluorescence data.
method Random finite set (RFS) based Bayesian approach.
result Gains 12% extra detection accuracy compared to MLSpike method.
Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations R, is extended to the first non-rectangular representations R=[2,1] and R=[3,1]. This increases chances that such factorization will take p…
PCA can detect a low-rank signal in spiked random matrix models, but not always optimally.
problem Understanding when PCA can detect a low-rank signal in the presence of noise.
method Le Cam's notion of contiguity, analysis of spiked Wishart ensemble, and non-spectral tests.
result PCA is sub-optimal for detection in non-Gaussian Wigner ensembles and certain negative spikes in Gaussian Wishart ensemble.
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a mi…
This paper analyzes generalization for linear models with spiked covariance structures.
problem Understanding the generalization performance of linear models with spiked covariance structures.
method Derives the generalization error for two simple models with spiked covariances using random matrix theory.
result The eigenvector and eigenvalue corresponding to the spike significantly influence the generalization error.
Accurate statistical models of neural spike responses can characterize the information carried by neural populations. But the limited samples of spike counts during recording usually result in model overfitting. Besides, current models assume spike counts to be Poisson-distributed, which ignores the fact that many neur…
Algorithm detects and estimates correlated signals in spiked matrices.
problem Detect and estimate correlated signals in spiked matrices.
method Proposes an efficient algorithm based on counting edge-decorated cycles.
result Algorithm succeeds under certain signal-to-noise ratio conditions.
Develops generic spike-and-slab priors for high-dimensional linear regression.
problem Bayesian high-dimensional linear regression challenges.
method Proposes a class of generic spike-and-slab priors and a unified framework for theoretical assessment.
result Achieves nearly-optimal posterior contraction rate and model selection consistency under general conditions.