In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tan…
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48 results for “rational entries”
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.
New algorithm recovers high-dimensional linear regression vectors without sparsity assumptions.
problem Efficiently recovering unknown vector β* from noisy linear observations in high dimensions.
method Proposes a polynomial-time algorithm based on LLL lattice basis reduction assuming rational entries with the same denominator.
result Algorithm successfully recovers β* for a large class of distributions and non-zero noise, even with small noise and one observation.
For a Morse map Novikov [11] has introduced an analog of Morse complex, defined over the ring $\ZZZ[[t]][t^{-1}]$ of integer Laurent power series. Novikov conjectured, that generically the matrix entries of the differentials in this complex are of the form , where grow at most exponenti…
Polynomial-time algorithm for inferring high-dimensional linear regression from a single sample.
problem Inferring an unknown feature vector from linear measurements in high dimensions without sparsity assumptions.
method Combining PSLQ integer relation detection and LLL lattice basis reduction algorithms.
result Polynomial-time recovery of from linear measurements , even with one sample.
Study of surface singularities and their deformations.
problem Characterize and classify quotient surface singularities and their deformations.
method Computation of resolutions, dual graphs, dimensions, Milnor numbers, and Milnor fibers.
result 6 pairs of minimal symplectic fillings are shown to be diffeomorphic.
Threadneedle is a multi-agent simulation framework, based on a full double entry book keeping implementation of the banking system's fundamental transactions. It is designed to serve as an experimental test bed for economic simulations that can explore the banking system's influence on the macro-economy under varying a…
The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on . It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a li…
Study of a generalized geometric Brownian motion with varying entry and exit rates.
problem Understanding the long-run behavior of economic systems with growth, volatility, entry, and exit.
method Generalized geometric Brownian motion framework with varying entry and exit rates, analyzing moments and survival probability.
result Optimal exit rate minimizes mean first-passage time, influencing system outcome.
Vectorizing string entries for data processing on tables: when are larger language models better?stat.ML
New research shows larger language models improve data processing for diverse entries.
problem Optimizing data processing for tables with diverse string entries.
method Analytical tasks on tables with varying language model sizes and a fuzzy join benchmark.
result Larger language models improve data processing for diverse entries, but fine-tuning is necessary.
We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless…
For any matrix A in R^(m x n) of rank ρ, we present a probability distribution over the entries of A (the element-wise leverage scores of equation (2)) that reveals the most influential entries in the matrix. From a theoretical perspective, we prove that sampling at most s = O ((m + n) ρ^2 ln (m + n)) entries of the ma…
Improved matrix completion for non-uniformly sampled data.
problem Estimating unobserved entries in a matrix with varying sampling probabilities.
method Developed entry-specific bounds for low-rank matrix completion under structured non-uniform sampling.
result Error bounds for each entry match minimax lower bounds under certain conditions.
Optimal Bubble Riding with Price-dependent Entry: a Mean Field Game of Controls with Common Noiseq-fin.MF
Study on optimal bubble riding with price-dependent entry times in a mean field game model.
problem Optimal bubble riding with price-dependent entry times.
method Mean field game of controls with common noise and random entry time, existence result obtained through discretization and limit analysis.
result Existence of equilibrium in the mean field game model.
Research shows how deepfakes can be used to manipulate accounting systems.
problem The vulnerability of CAATs to adversarial attacks.
method Developed a thread model to camouflage anomalies, used adversarial autoencoder neural networks to learn latent factors, demonstrated misuse of model to generate misleading entries.
result Adversarial autoencoder neural networks can learn and manipulate accounting data to deceive CAATs.
We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More s…
To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…
Paper improves tensor completion by reducing sample entries needed.
problem Reducing the number of required sample entries for tensor completion.
method Utilizes multi-rank and unitary transformation in tensor singular value decomposition.
result Provides a bound on the number of required sample entries for tensor completion.
Reducing barriers to entry in large-scale ML markets, study shows multi-objective learning can lower data requirements.
problem Barriers to entry in emerging markets for large-scale machine learning models.
method Defined a multi-objective high-dimensional regression framework to study reputational damage and data requirements.
result The number of data points needed for a new company to enter the market can be significantly smaller than the incumbent company's dataset size.
Develops a method to complete binary matrices using all types of observed entries.
problem Completing binary matrices from partial observations.
method Combines risks from Davenport et al. (2014) and Hsieh et al. (2015) to use all types of entries.
result Improves matrix completion performance by using all types of entries.
A new tensor completion method handles missing data with missing not at random entries.
problem Handling missing data in tensors where the probability of observation depends on other entries.
method Estimate propensities using convex relaxation, then use higher-order SVD with inverse propensities weights.
result Finite-sample error bounds on the completed tensor are provided.
Numerical groupsmath.GR
Groups of matrices with integer-like entries are studied.
problem Characterizing groups of matrices with algebraic integer entries.
method Analyzing traces and subgroups of matrices in number fields.
result Irreducible or completely reducible subgroups with algebraic integer traces are numerical.
Study on market entry timing in stock liquidation with trading constraints.
problem Optimal timing of market entry and exit in portfolio liquidation with trading restrictions.
method Mean-field game approach to model -player and mean-field games of optimal portfolio liquidation.
result Existence of unique equilibrium in both mean-field and -player games.
The completion of low rank matrices from few entries is a task with many practical applications. We consider here two aspects of this problem: detectability, i.e. the ability to estimate the rank reliably from the fewest possible random entries, and performance in achieving small reconstruction error. We propose a …
Estimates low-rank distributional matrices from incomplete samples.
problem Matrix completion for distributional entries with limited observed data.
method Kernel mean embeddings, Tucker rank, functional unfolding operators.
result Effective estimator for distributional matrix completion established.
Scalable and robust TR decomposition for large-scale data with missing entries and outliers.
problem Handling large-scale tensor data with missing entries and outliers.
method Auto-weighted steepest descent method for missing entries and outliers identification, FGMC and RStS strategies.
result Outperforms existing TR decomposition methods in the presence of outliers and runs faster than robust tensor completion algorithms.
In this paper, we consider the streaming memory-limited matrix completion problem when the observed entries are noisy versions of a small random fraction of the original entries. We are interested in scenarios where the matrix size is very large so the matrix is very hard to store and manipulate. Here, columns of the o…
We give an algorithm for completing an order- symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic result…
The covariance matrix of a -dimensional random variable is a fundamental quantity in data analysis. Given i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of operations. When are large, this computation may be prohibitively slow. Moreover, …
Paper analyzes SSC for data with missing entries, improving performance.
problem Theoretical analysis of SSC with missing data entries.
method Analyzes theoretical guarantees for SSC with incomplete data, projecting zero-filled data onto observation pattern.
result Improves performance of SSC with incomplete data by projecting zero-filled data onto observation pattern.
The paper completes matrices from non-uniformly sampled entries, especially when columns are randomly selected and fully observed.
problem Matrix completion from non-uniformly sampled entries, including fully and partially observed columns.
method First, recover the column space from fully observed columns. Then, for each partially observed column, find a vector in the recovered column space with the observed entries. For low-rank matrices, recover them from entries.
result The algorithm can exactly recover a low-rank matrix from merely entries.
A new method for streaming PCA provides confidence intervals for eigenvector entries.
problem Uncertainty quantification for individual entries in streaming PCA.
method Oja's algorithm, Bernstein-type concentration bound, Central Limit Theorem, subsampling algorithm.
result Sharp concentration bound and Central Limit Theorem for streaming PCA entries.
BIND removes background noise from binary matrices, improving detection accuracy and fairness.
problem Real data often violates the i.i.d assumption for binary matrix entries, leading to inaccurate detection.
method BIND optimizes detection by estimating row- and column-wise mixture distributions and eliminating background noise.
result BIND effectively removes background noise and increases detection accuracy and fairness.
PACE-GGM uses Gaussian mechanism for private covariance estimation.
problem Private estimation of covariance matrices in high dimensions.
method Data-adaptive selection of entries, Gaussian mechanism, maximum-entropy reconstruction.
result Consistent improvements in estimation error compared to Gaussian mechanism and baselines.
We study low rank matrix and tensor completion and propose novel algorithms that employ adaptive sampling schemes to obtain strong performance guarantees. Our algorithms exploit adaptivity to identify entries that are highly informative for learning the column space of the matrix (tensor) and consequently, our results …
Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn/|E|)^0.5 . Further, if r=O(1), M can be reconstructed ex…
New spectral methods improve matrix estimation in RL with low-rank structure.
problem Estimating matrices with low-rank structure in reinforcement learning.
method Spectral-based matrix estimation approaches.
result Spectral methods efficiently recover singular subspaces and minimize entry-wise error.
TATD predicts missing entries in time-evolving tensors by exploiting temporal dependency and sparsity.
problem Predict missing entries in time-evolving tensors with temporal dependency and sparsity issues.
method TATD (Time-Aware Tensor Decomposition) integrates temporal dependency and time-varying sparsity through a smoothing regularization with Gaussian kernel and alternating optimization.
result TATD achieves state-of-the-art accuracy for decomposing temporal tensors.
A new method detects and compacts saturated entries in antisparse coding.
problem Efficiently solving antisparse coding problems with -norm penalties.
method Safe squeezing methodology to detect and compact saturated entries, reducing problem dimensionality.
result The method accelerates the computation of antisparse representation by detecting and compacting saturated entries.
We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determina…
LEMN improves memory networks for learning from streaming data.
problem Efficiently processing large data streams in memory-augmented neural networks.
method LEMN uses a RNN-based retention agent to learn and replace less important memory entries based on their importance and historical context.
result LEMN achieves significant improvements over existing methods in learning from streaming data.
New algorithm optimizes positions of CountSketch non-zero entries for better data compression.
problem Optimizing positions of CountSketch non-zero entries for better data compression.
method Learning algorithm that optimizes both values and positions of CountSketch non-zero entries.
result Improves accuracy for low rank approximation and other problems like k-means clustering.
We consider the problem of reconstructing a low rank matrix from a subset of its entries and analyze two variants of the so-called Alternating Minimization algorithm, which has been proposed in the past. We establish that when the underlying matrix has rank , has positive bounded entries, and the graph $\mathcal{G…
Study heavy-tailed weights' impact on neural network's spectral distribution.
problem Analyzing spectral distribution of conjugate kernel matrices with heavy-tailed weights.
method Computed limiting eigenvalue distribution through moments, considering heavy-tailed distributions and nonlinear activation functions.
result Heavy-tailed weights induce strong correlations, leading to fundamentally different spectral behavior.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
We improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. 1) In compressed sensing, we show that if the m \times n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tr…
Method uses optimal transport to complete distributional matrices.
problem Matrix completion for distributional data.
method Nearest neighbors in Wasserstein space.
result Method recovers distributions in Wasserstein metric.
New approach identifies hidden degrees of freedom in M-theory.
problem Identifying hidden degrees of freedom at ADE-singularities in M-theory.
method Real equivariant cohomotopy on superspaces, using Elmendorf's theorem.
result Reveals ADE-singularities as part of super-cocycles on 11d super-spacetime.