HSNLD solves robust Hankel recovery efficiently and robustly.
problem Robust Hankel recovery of sparse outliers and missing entries.
method Hankel Structured Newton-Like Descent (HSNLD) algorithm.
result HSNLD achieves linear convergence independent of the condition number.
An algorithm finds a compact Hankel submatrix for spectral learning.
problem Efficiently computing SVD for large Hankel matrices in spectral learning.
method Maximal bipartite matching algorithm to select rows and columns of Hankel matrix.
result Compact Hankel submatrix with full structural rank.
The paper tackles system identification via Hankel nuclear norm regularization, improving estimation rates and singular value gaps.
problem Identifying low-order linear systems from limited data.
method Hankel nuclear norm regularization to encourage low-rankness of the Hankel matrix.
result Hankel regularization enables optimal system recovery with fewer observations and better estimation rates.
New model mimics neural next item recommendation using Hankel matrices.
problem Next item recommendation efficiency and structural knowledge capture.
method Tensor factorization with Hankel matrix representation.
result Model performs competitively with neural networks but is simpler.
Deep learning improves MRI image reconstruction from sparse k-space data.
problem Accelerated MRI imaging with limited k-space data.
method Data-driven deep learning using convolutional neural networks and Hankel matrix decomposition.
result Deep learning consistently outperforms existing image-domain methods in k-space MRI reconstruction.
Spectral regularization simplifies sequence models by focusing on grammatical simplicity.
problem Sequence modeling challenges in learning tasks.
method Introduces spectral regularization based on Hankel matrices and trace norm, addressing bi-infinite matrices with an unbiased estimator.
result Demonstrates spectral regularization's potential benefits on Tomita grammars.
The paper reviews Hankel low-rank methods for time series analysis and forecasting.
problem Developing efficient methods for time series analysis and forecasting.
method Hankel low-rank approximation and completion techniques.
result Discussion of methods and challenges in obtaining optimal solutions.
Paper tackles missing value imputation in time series forecasting.
problem Missing value imputation in time series analysis.
method Low-rank matrix completion with Hankel matrices and nuclear norm relaxation.
result Proper weighting scheme is crucial for known observations.
Novel factorization for low-rank matrices in subspaces, improving efficiency.
problem Learning low-rank matrices constrained to subspaces.
method Riemannian manifold optimization with conjugate gradient and trust-region algorithms.
result Efficient algorithms for structured low-rank matrix learning.
New method controls linear systems with adversarial disturbances.
problem Controlling linear dynamical systems under adversarial conditions.
method Novel convex relaxation using spectral filters from Hankel matrix eigenvectors.
result Polylogarithmic running time improvement over prior methods.
Paper speeds up GP inference by reducing precision matrix computation.
problem High computational complexity in computing kernel precision matrices.
method Splitting precision matrix into Hankel-Toeplitz matrices and computing only unique entries.
result Precision matrix computation reduced from O(NM2) to O(NM). Noise-robust Koopman operator framework for control with improved stability and performance.
problem Developing a stable and noise-robust Koopman operator for control tasks.
method Proposes a learning framework using Hankel matrix and neural network approximations for system dynamics, ensuring long-term stability and noise robustness.
result Demonstrates improved model performance and noise robustness in control tasks compared to existing methods.
This paper proposes a method to select bases for spectral learning of PSRs using model entropy.
problem Learning PSR models with limited data and computational resources.
method Adopting model entropy to select columns for spectral learning of PSRs.
result The proposed method can effectively select bases for spectral learning of PSRs.
The paper tackles estimation of hidden state LTI systems of unknown order.
problem Estimation of Markov parameters and minimal realization of unknown order LTI systems.
method Hankel penalized least square estimator, Ho-Kalman algorithm, and a combined algorithm.
result Statistical guarantees for estimation error, rank recovery, and sample complexity.
The paper studies the problem of recovering a spectrally sparse object from a small number of time domain samples. Specifically, the object of interest with ambient dimension n is assumed to be a mixture of r complex multi-dimensional sinusoids, while the underlying frequencies can assume any value in the unit disk…
New nonconvex methods improve SysID efficiency and accuracy.
problem Efficiently identify low-order linear systems from limited data.
method Proposes two nonconvex reformulations of Hankel-rank minimization for SysID.
result Nonconvex methods achieve lower statistical error rates and sample complexities.
The paper explores the problem of \emph{spectral compressed sensing}, which aims to recover a spectrally sparse signal from a small random subset of its n time domain samples. The signal of interest is assumed to be a superposition of r multi-dimensional complex sinusoids, while the underlying frequencies can assum…
Algorithm uses matrix estimation to impute and forecast time series data.
problem Impute and forecast time series data with missing values and noise.
method Transform time series into a matrix, use matrix estimation for missing values and de-noise, perform linear regression for predictions.
result Established a rigorous link between time series analysis and matrix estimation, providing finite sample analysis and asymptotic consistency.
A k-space deep learning method corrects EPI ghost artifacts without a reference scan.
problem Nyquist ghost artifacts in EPI MRI due to phase mismatch between even and odd echoes.
method Structured low-rank Hankel matrix approaches combined with data-driven Hankel matrix decomposition and deep convolutional neural networks.
result The proposed k-space deep learning method outperforms existing methods in image quality and computing time.
This paper explores robust recovery of a superposition of R distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of 2N−1 dimensional and R<<2N−1. This framework covers a large class of signals arising from real applications in biology, automation,…
Improved modeling of chaotic systems using time-delay embeddings and Frenet-Serret frame.
problem Identifying effective coordinate systems for nonlinear dynamical systems.
method Developed a new algorithm to identify more stable and accurate models from less data, leveraging the connection between HAVOK and Frenet-Serret frame.
result The sub- and super-diagonal entries of the linear model correspond to intrinsic curvatures in Frenet-Serret frame.
Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. Bu…
Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on ℓ2-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t classical parametric met…
Efficient algorithm predicts discrete-time linear systems using spectral filtering.
problem Online prediction of discrete-time linear dynamical systems.
method Improper learning to convexify the loss functions, then using spectral filtering.
result Near-optimal regret and sample complexity guarantees for agnostic learning.
Proposes BHT-ARIMA for forecasting multiple short time series.
problem Forecasting multiple short time series with mutual correlations.
method Block Hankel tensors, Tucker decomposition, generalized tensor ARIMA.
result Improves forecasting accuracy and reduces computational cost.
HOPE improves SSMs for long-memory tasks with robust initialization and training.
problem Improving state-space models for long-memory tasks with robust initialization and training.
method Developed a new parameterization scheme called HOPE using Hankel operators and Markov parameters.
result HOPE improves SSMs' performance on Long-Range Arena tasks and demonstrates non-decaying memory.
In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical the…
Algorithm learns graph operator from sparse space-time samples.
problem Learning time-varying graph signals from partial observations.
method Non-convex IRLS algorithm for low-rank matrix completion.
result No more than O(rn log(nT)) space-time samples needed for accurate recovery.
Reservoir computing's success depends on mapping different input time series to separable states.
problem Quantifying the ability of random linear reservoirs to map different input time series.
method Mathematical framework using spectral properties of the connectivity matrix.
result Separation capacity is fully characterized by the spectral properties of the connectivity matrix.
The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, w…
Paper connects WFA and 2-RNNs, offering a new learning algorithm.
problem Expressiveness and learning of recurrent neural networks.
method Spectral learning algorithm for linear 2-RNNs.
result Provable learning algorithm for linear 2-RNNs.
Optimal joint separation condition for radar and communications channels in dual-blind deconvolution.
problem Recovering information from overlaid radar and communications signals with unknown channels.
method Extremal functions from Beurling-Selberg interpolation theory for joint separation, nuclear norm minimization for matrix retrieval, and MUSIC for parameter estimation.
result Guaranteed well-conditioned Vandermonde matrix for MUSIC, validating theoretical findings.
Constructs new topological theories in 2D not fitting standard axioms.
problem Developing new topological theories in 2D that don't conform to traditional axioms.
method Universal construction by Blanchet et al., Kronecker's characterization, field extension, Hankel matrices, Schur polynomials, and foam evaluation.
result Introduction of non-multiplicative theories and classification over finite-dimensional state spaces.
This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this trade-off is determined by one crucial design parameter. The main contribution is a methodology to app…
A neural network, IHT-Net, improves DOA estimation with sparse arrays.
problem Single-snapshot DOA estimation with sparse arrays in dynamic settings.
method IHT-inspired neural network with recurrent neural network and autoencoders.
result IHT-Net achieves faster convergence and higher accuracy in DOA estimation.
Algorithm learns linear systems from partial observations with near-optimal rate.
problem Identifying linear dynamical systems from partial observations, especially those with long-term memory.
method Multi-scale low-rank approximation using SVD on Hankel matrices of increasing sizes, combined with Fourier domain concentration bounds.
result Near-optimal rate of $\widetilde O\left(\sqrt\frac{d}{T}
ight)$ in H2 error, with logarithmic dependence on memory length. Let γ:I→Rn be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of γ at γ(t) consists of a frame e1(t),…,en(t) and generalized curvature values κ1(t),…,κn−1(t). Associated with each point of γ there are also local singular vecto…
We study power expansions of the characteristic function of a linear operator A in a p∣q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. The…
We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of s…
Deep convolution framelets improve deep learning for inverse problems.
problem Improving deep learning performance in inverse problems.
method Developed convolution framelets for signal representation, combined with deep neural networks.
result Deep convolution framelets achieve perfect reconstruction (PR) and outperform existing architectures.
This paper addresses network anomography, that is, the problem of inferring network-level anomalies from indirect link measurements. This problem is cast as a low-rank subspace tracking problem for normal flows under incomplete observations, and an outlier detection problem for abnormal flows. Since traffic data is lar…
The paper introduces a diagnostic method to detect grokking transitions in models before test accuracy improves.
problem Detecting the transition from training to generalization in machine learning models.
method Summarize task-dependent observables as empirical distributions, map them to Wasserstein/quantile coordinates, and analyze using Hankel dynamic mode decomposition.
result The diagnostic method achieves AUROC \(\approx\) 0.93 for grokking-vs-non-grokking discrimination at the run level.
DeepTMR reorders matrices without prior knowledge of structural patterns.
problem Matrix reordering without prior structural knowledge.
method DeepTMR uses a neural network to automatically extract features and reorder matrices.
result Trained network produces denoised mean matrix for visualization.
Paper speeds up matrix multiplication on Intel PIII using SIMD.
problem Efficiently multiplying large matrices for faster algorithm performance.
method Implemented matrix-matrix multiply using Intel Pentium SIMD architecture.
result Average performance 2.09 times faster than public domain routines.
The paper constructs Goeritz matrices from Dehn colorings.
problem Constructing Goeritz matrices from Dehn colorings.
method Purely algebraic construction of Goeritz matrices from Dehn coloring matrices for prime knot diagrams.
result A new method to construct Goeritz matrices from Dehn colorings.
New matrix reveals cluster info in sparse directed graphs.
problem Analyzing cluster information in directed graphs.
method Proposed complex non-backtracking matrix integrating Hermitian adjacency matrix and non-backtracking matrix properties.
result The complex non-backtracking matrix holds cluster information, especially for sparse directed graphs.
The CN matrix of a pure braid projection is characterized and applied.
problem Understanding the structure of CN matrices for braid projections.
method Discussion and characterization of patterns and specific matrices.
result Characterization of CN matrix of a pure 6-braid projection and related matrices.
Characterizes the OU matrix for up to 5 strands in braids.
problem Understanding the structure of braid diagrams through their matrices.
method Characterization of the OU matrix for up to 5 strands in braids.
result Standard form of the OU matrix for general braids of up to 5 strands is given and characterized.