The End Of Knowledge - Vault 6/80 - CVPR - 2022 - Learning Mixtures of Linear Dynamical Systems

graph LR classDef linear fill:#f9d4d4, font-weight:bold, font-size:14px classDef estimation fill:#d4f9d4, font-weight:bold, font-size:14px classDef methods fill:#d4d4f9, font-weight:bold, font-size:14px classDef analysis fill:#f9f9d4, font-weight:bold, font-size:14px A[Learning Mixtures of
Linear Dynamical Systems] --> B[Linear
Systems] A --> C[Model
Estimation] A --> D[Methods] A --> E[Analysis] B --> B1[Linear models
evolving
over time. 1] B --> B2[Multiple models
generating unlabeled
data. 2] B --> B3[Grouping similar
data
trajectories. 3] B --> B4[Assigning new
data to
clusters. 4] B --> B5[Random noise
following normal
distribution. 14] B --> B6[Future states
independent of
initial conditions. 9] C --> C1[Inferring model
parameters from
data. 5] C --> C2[Identifying relevant
low-dimensional
subspaces. 6] C --> C3[Coarse estimation
then refinement
steps. 7] C --> C4[Samples needed for
accurate
estimation. 10] C --> C5[Constraints ensuring
model
stability. 15] C --> C6[Minimum differences
for model
distinction. 16] D --> D1[Eigenvalue methods
for dimensionality
reduction. 11] D --> D2[Minimizing squared
prediction
errors. 12] D --> D3[Maximizing observed
data
probability. 13] D --> D4[Projecting data
to lower
dimensions. 18] D --> D5[Bounds on subspace
differences. 21] D --> D6[Operation for covariance
calculations. 26] E --> E1[Robust technique
for handling
outliers. 19] E --> E2[Estimators with tighter
error
bounds. 20] E --> E3[Combining multiple
high-probability
events. 23] E --> E4[Extending results
from finite to
continuous. 24] E --> E5[Bounds on eigenvalue
differences. 25] E --> E6[Model order doesnt
affect
mixture. 27] class A,B,B1,B2,B3,B4,B5,B6 linear class C,C1,C2,C3,C4,C5,C6 estimation class D,D1,D2,D3,D4,D5,D6 methods class E,E1,E2,E3,E4,E5,E6 analysis

Resume:

1.- Linear Dynamical Systems (LDS): Mathematical models describing systems that evolve over time according to linear equations, with state transitions and noise.

2.- Mixed LDS: Multiple LDS models generating unlabeled trajectories, with unknown labels for each trajectory.

3.- Clustering: Grouping similar trajectories together, assuming they were generated by the same LDS model.

4.- Classification: Assigning new trajectories to existing clusters based on their likelihood of being generated by each model.

5.- Model estimation: Inferring parameters of LDS models (state transition matrices, noise covariance matrices) from grouped trajectories.

6.- Subspace estimation: Identifying low-dimensional subspaces containing relevant information about the LDS models to reduce dimensionality.

7.- Two-stage algorithm: Coarse estimation followed by refinement, involving clustering, classification, and model estimation steps.

8.- Autocovariance matrices: Statistical descriptors of LDS models, used for comparing and distinguishing between different models.

9.- Mixing property: Characteristic of LDS where future states become increasingly independent of initial conditions over time.

10.- Sample complexity: Number of samples required to achieve accurate model estimation with high probability.

11.- Spectral methods: Techniques using eigenvalue decomposition for subspace estimation and dimensionality reduction.

12.- Least squares estimation: Method for estimating LDS parameters by minimizing squared differences between predicted and observed states.

13.- Maximum likelihood estimation: Technique for estimating model parameters by maximizing the probability of observed data.

14.- Gaussian noise: Random perturbations in LDS models assumed to follow a normal distribution.

15.- Stability conditions: Constraints on LDS parameters ensuring the system doesn't diverge over time.

16.- Separation conditions: Minimum differences between LDS models to ensure they can be distinguished.

17.- Short trajectories: Sample paths from LDS models with lengths much smaller than the state dimension.

18.- Dimension reduction: Techniques to project high-dimensional data onto lower-dimensional subspaces while preserving important information.

19.- Median-of-means estimator: Robust statistical technique used in the clustering algorithm to handle outliers.

20.- Self-normalized concentration: Property of certain statistical estimators that allows for tighter error bounds.

21.- Davis-Kahan theorem: Result bounding differences between subspaces of perturbed matrices, used in subspace estimation analysis.

22.- Hanson-Wright inequality: Concentration result for quadratic forms of random vectors, used in covariance estimation.

23.- Union bound: Probabilistic tool for combining multiple high-probability events, used throughout the theoretical analysis.

24.- Covering arguments: Technique for extending results from finite sets to continuous spaces, used in concentration proofs.

25.- Weyl's inequality: Result bounding differences in eigenvalues of perturbed matrices, used in spectral analysis.

26.- Kronecker product: Operation on matrices used in covariance calculations for vectorized LDS states.

27.- Permutation invariance: Property where the ordering of LDS models doesn't affect the overall mixture.

28.- Modular algorithm design: Structuring algorithms as separate components that can be modified or replaced independently.

29.- End-to-end guarantees: Theoretical results ensuring the entire algorithm pipeline achieves desired accuracy with high probability.

30.- Polynomial sample complexity: Bounds on required sample sizes that grow as polynomials in problem parameters.

Knowledge Vault built byDavid Vivancos 2024