An official website of the United States government
This article introduces an isometric manifold embedding data-driven paradigm designed to enable model-free simulations with noisy data sampled from a constitutive manifold. The proposed data-driven approach iterates between a global optimization problem that seeks admissible solutions for the balance principle and a local optimization problem that finds the closest point projection of the Euclidean space that isometrically embeds a nonlinear constitutive manifold. To de-noise the database, a geometric autoencoder is introduced such that the encoder first learns to create an approximated embedding that maps the underlying low-dimensional structure of the high-dimensional constitutive manifold onto a flattened manifold with less curvature. We then obtain the noise-free constitutive responses by projecting data onto a denoised latent space that is completely flat by assuming that the noise and the underlying constitutive signal are orthogonal to each other. Consequently, a projection from the conservative manifold onto this de-noised constitutive latent space enables us to complete the local optimization step of the data-driven paradigm. Finally, to decode the data expressed in the latent space without reintroducing noise, we impose a set of isometry constraints while training the autoencoder such that the nonlinear mapping from the latent space to the reconstructed constituent manifold is distance-preserving. Numerical examples are used to both validate the implementation and demonstrate the accuracy, robustness, and limitations of the proposed paradigm.
more »
« less
Identification of Error-in-Variables Switched Systems using a Riemannian Embedding
This paper considers the problem of error in variables identification for switched affine models. Since it is well known that this problem is generically NP hard, several relaxations have been proposed in the literature. However, while these approaches work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. To address this difficulty, we propose a computationally efficient alternative, based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric there to perform the identification. Our main result shows that, under dwell-time assumptions, the proposed algorithm is convergent, in the sense that it is guaranteed to identify the system for suitably low noise. In scenarios with larger noise levels, we provide experimental results showing that the proposed method outperforms existing ones. The paper concludes by illustrating these results with academic examples and a non-trivial application: action video segmentation.
more »
« less
- PAR ID:
- 10447155
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This paper addresses the problem of identification of error in variables switched linear models from experimental input/output data. This problem is known to be generically NP hard and thus computationally expensive to solve. To address this difficulty, several relaxations have been proposed in the past few years. While solvable in polynomial time these (convex) relaxations tend to scale poorly with the number of points and number/order of the subsystems, effectively limiting their applicability to scenarios with relatively small number of data points. To address this difficulty, in this paper we propose an efficient method that only requires performing (number of subsystems) singular value decompositions of matrices whose size is independent of the number of points. The underlying idea is to obtain a sum-of-squares polynomial approximation of the support of each subsystem one-at-a-time, and use these polynomials to segment the data into sets, each generated by a single subsystem. As shown in the paper, exploiting ideas from Christoffel's functions allows for finding these polynomial approximations simply by performing SVDs. The parameters of each subsystem can then be identified from the segmented data using existing error-in-variables (EIV) techniques.more » « less
-
This paper studies the data-driven reconstruction of firing rate dynamics of brain activity described by linear-threshold network models. Identifying the system parameters directly leads to a large number of variables and a highly non-convex objective function. Instead, our approach introduces a novel reformulation that incorporates biological organizational features and turns the identification problem into a scalar variable optimization of a discontinuous, non-convex objective function. We prove that the minimizer of the objective function is unique and establish that the solution of the optimization problem leads to the identification of all the desired system parameters. These results are the basis to introduce an algorithm to find the optimizer by searching the different regions corresponding to the domain of definition of the objective function. To deal with measurement noise in sampled data, we propose a modification of the original algorithm whose identification error is linearly bounded by the magnitude of the measurement noise. We demonstrate the effectiveness of the proposed algorithms through simulations on synthetic and experimental data.more » « less
-
Robust PCA is a widely used statistical procedure to recover an underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices and proposes two algorithms based on manifold optimization. It is shown that, with a properly designed initialization, the proposed algorithms are guaranteed to converge to the underlying lowrank matrix linearly. Compared with a previous work based on the factorization of low-rank matrices Yi et al. (2016), the proposed algorithms reduce the dependence on the condition number of the underlying low-rank matrix theoretically. Simulations and real data examples con rm the competitive performance of our method.more » « less
-
Hybrid beamforming (HBF) is a key enabler for millimeter-wave (mmWave) communications systems, but HBF optimizations are often non-convex and of large dimension. In this paper, we propose an efficient deep unfolding-based HBF scheme, referred to as ManNet-HBF, that approximately maximizes the system spectral efficiency (SE). It first factorizes the optimal digital beamformer into analog and digital terms, and then reformulates the resultant matrix factorization problem as an equivalent maximum-likelihood problem, whose analog beamforming solution is vectorized and estimated efficiently with ManNet, a lightweight deep neural network. Numerical results verify that the proposed ManNet-HBF approach has near-optimal performance comparable to or better than conventional model-based counterparts, with very low complexity and a fast run time. For example, in a simulation with 128 transmit antennas, it attains 98.62% the SE of the Riemannian manifold scheme but 13250 times faster.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2023.3275973
