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.
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.
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.
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…
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.
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.
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.
We present a solution to scale spectral algorithms for learning sequence functions. We are interested in the case where these functions are sparse (that is, for most sequences they return 0). Spectral algorithms reduce the learning problem to the task of computing an SVD decomposition over a special type of matrix call…
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.
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.
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.
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.
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.
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.
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.
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…
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.
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). 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.
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.
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…
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,…
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. 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.
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…
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…
Predictive State Representations (PSRs) are powerful techniques for modelling dynamical systems, which represent a state as a vector of predictions about future observable events (tests). In PSRs, one of the fundamental problems is the learning of the PSR model of the underlying system. Recently, spectral methods have …
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…
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 consider the problem of learning a low-rank matrix, constrained to lie in a linear subspace, and introduce a novel factorization for modeling such matrices. A salient feature of the proposed factorization scheme is it decouples the low-rank and the structural constraints onto separate factors. We formulate the optim…
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.
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.
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…
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.
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…
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…
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…
The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.
problem Understanding curvature tensors and their minimal norm.
method Analyzing minimal norm tensors for third and fourth covariant tensors, including Riemannian curvature and divergence.
result Weyl tensor and Cotten tensor are identified as minimal norm tensors of Riemannian curvature and divergence tensors, respectively.
A new tree method for tensor data improves regression accuracy.
problem Efficiently modeling tensor data for regression problems.
method Scalar-output regression tree models for scalar-on-tensor problems, and tensor-on-tensor problems using additive tree ensemble approaches.
result The tensor-input tree (TT) method outperforms tensor-input GP models in efficiency and accuracy.
Curvature tensors can always be matched to a metric tensor under certain conditions.
problem Sectionally positive curvature tensors and their relationship to metric tensors.
method Existence and uniqueness of a metric tensor gab such that Rabcdgbd=gacλ. result A metric tensor gab can be found for sectionally positive curvature tensors, and it is unique up to a constant factor. Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
A new tensor decomposition method that minimizes KL divergence.
problem Tensor reconstruction accuracy.
method Legendre decomposition, based on information geometry.
result Minimizes KL divergence and improves tensor reconstruction accuracy.
A Matlab toolbox for tensor operations based on t-product.
problem Extending matrix operations to tensors.
method Developed a Matlab toolbox implementing tensor operations based on t-product.
result Implemented several tensor operations including SVD, spectral norm, and nuclear norm.
Paper improves tensor completion using unitary transforms.
problem Robust tensor completion for various datasets.
method Transformed tensor SVD with unitary matrices.
result Recovered images have better PSNR than traditional methods.
Compatible tensors form a special Jordan algebra.
problem Understanding the algebraic structure of compatible tensors.
method Proving tensors form a Jordan algebra through symmetrized product properties.
result Riemann, Weyl, and curvature compatible tensors form a special Jordan algebra.
Paper optimizes tensor deflation for non-orthogonal signals.
problem Recovering low-rank signals from noisy tensors with correlated components.
method Developed an asymptotic analysis and optimized deflation procedure using random tensor theory.
result Proposed an efficient tensor deflation algorithm that optimizes a parameter introduced in the deflation mechanism.
Paper proposes a new method for exact recovery in robust tensor principal component analysis.
problem Exact recovery of low-rank and sparse components in tensors.
method Proposes a new method based on tensor-tensor product and t-SVD to solve a convex optimization problem.
result Exact recovery achieved in a deterministic fashion without randomness assumptions.
Efficiently decomposes large tensors using stochastic gradients.
problem Efficiently decomposing large tensors for multiway data analysis.
method Stochastic gradients computed via MTTKRP kernel for efficient computation.
result Advantages and scalability demonstrated for large-scale problems.