Learning structured linear dynamical systems from missing observations

arXiv:2610.11869 · stat.ML, cs.LG, cs.SY, eess.SY, math.OC, math.ST, stat.TH · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Learning structured linear dynamical systems from missing observations".

Jane: As a fastidious and diligent AI researcher,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: We’re building on the idea that we can estimate a structured linear dynamical system matrix even with missing observations, and this paper lays out the formal summary of their approach to "Learning structured linear dynamical systems from missing observations."

Jane: Basically, they frame it as minimizing a bias-corrected least squares problem under a convex set K where only some entries are observed at each time point. The core idea is that the estimator you propose could be non-convex because of that constraint set K, which is why they reference the work of Loh and Wainwright (two thousand eleven) <ref:2610.11869#pg2,of Loh and Wainwright (2011>.

Lu: Their main theoretical contribution is deriving estimation error bounds in the Frobenius norm, and these bounds depend directly on how complex the tangent cone of K at any point B in K is, quantified by Talagrand’s gamma one and gamma two functionals.

Meng: So it’s not just a simple formula; the statistical error depends on the geometry of the set K itself, which is a deep connection between estimation theory and geometric complexity. How does that translate to something we can actually compute?

Lalam: They also show that they can achieve meaningful recovery of A* under specific structural assumptions. They apply their general theory to three specific settings: when K is a subspace, when K is the set of bi-isotonic matrices, and when K is the set of matrices whose rows are formed by sampling Lipschitz functions.

Tom: That’s a lot of specific cases they cover for this problem. It suggests that the general framework has enough flexibility to handle several common structures in dynamical systems, not just one simple case.

Jane: Yes, and their results suggest that meaningful recovery of the transition matrix is actually possible for values of T much smaller than what is required in the unconstrained case, and for a sub-sampling probability p that goes to zero slowly.

Lu: They show that this recovery happens when T scales sub-quadratically with the system dimension n, specifically T much smaller than n squared, and for p being less than one <ref:2610.11869#pg2>.

The paper's summary: Tom: Moving on to what they suggest as improvements over prior work, this paper really focuses on pushing the limits of what’s possible in this missing data scenario.

Jane: One major improvement is establishing a linear convergence rate for Projected Gradient Descent under high probability, which is Theorem two <ref:2610.11869#pg2>. This means the iterative learning process itself gets better and better at finding the true system matrix A*.

Meng: Linear convergence is definitely a big step up from what you might see with simpler first-order methods that might only show sublinear rates. It makes the learning algorithm much more reliable when we are running these kinds of estimations.

Lalam: The additive error epsilon in that linear convergence bound is also quite informative, showing exactly how the error shrinks over time and includes a term p (one/delta) sqrt T which relates to the confidence level of the result <ref:2610.11869#pg1>.

Lu: They also show that they can achieve meaningful recovery even when T is small relative to n squared and p is small, which, as we discussed, points toward a more accessible regime for identification problems <ref:2610.11869#pg2>.

Tom: And they also offer a concrete way to handle missing data imputation strategies, specifically by using imputing the missing entries by zero. They provide provable error rates for structured A* recovery in this scenario when you use that specific zero-imputation strategy.

Jane: So it gives us a reliable method not just theoretically, but with a concrete strategy for handling the missing data aspect of this problem. It moves it from being purely theoretical to being more practically applicable.

The paper's improvements: Tom: We’re wrapping up our discussion on "Learning structured linear dynamical systems from missing observations." To summarize, the core finding is that under appropriate conditions on T and p, we can provably recover the transition matrix A* using a bias-corrected least squares problem.

Jane: The results show that this estimation error is controlled by structural complexity measures of K, like gamma one and gamma two, which tell us how complex the constraint set is at the true matrix A*.

Lu: So, the overall implication is that meaningful recovery of A* is achievable under conditions where T scales sub-quadratically with n and p vanishes slowly.

Meng: From an engineering standpoint, this means we have a theoretical guarantee for using PGD to estimate systems when data is incomplete, provided we stick to the required trajectory length T and probability p.

Lalam: And for our future work, they point toward analyzing how these structural complexities play out in more advanced settings, which could lead to better ways to improve AI systems by understanding the underlying system dynamics more deeply.

Tom: That’s all we have time for today on this paper, but it’s a solid piece of work showing that structured A* recovery is definitely possible even with missing observations.

Conclusion: Tom: So we’ve been deep into "Learning structured linear dynamical systems from missing observations," and basically, this paper shows you can actually reconstruct the underlying system matrix A* even when you have missing data points along the way.

Jane: It frames it as solving a bias-corrected least squares problem under some constraint set K, which is really interesting because that constraint set K determines how complex the estimation error gets.

Lu: The authors use Talagrand functionals, specifically gamma one and gamma two, to measure that complexity of the constraint set at any given point in the matrix space.

Meng: So it’s not just a simple formula for an estimator; it shows that the statistical error depends on how geometrically complex that constraint set K is around our true system A*.

Lalam: And they prove linear convergence using Projected Gradient Descent, meaning their iterative algorithm gets predictably better and better at finding the right matrix, with a specific learning rate gamma they derived.

Tom: That’s solid. The big win here is that they show recovery is possible even when the trajectory length T isn't huge—we're talking sub-quadratic scaling with the system size n, as long as p doesn't vanish too fast.

Jane: It changes things because it moves this identification problem from needing massive amounts of complete data to being solvable with sparser, incomplete information under specific mathematical constraints.

Lu: They look at three specific settings—a subspace, bi-isotonic matrices, and Lipschitz regression—and show the general framework handles all of them.

Meng: From an engineering standpoint, that means we can design practical algorithms for real-world systems where sensors drop out or data packets are lost without having to wait for perfect information.

Lalam: And they even provide a strategy: if you impute those missing entries by setting them to zero, they give you provable error rates for recovering A*.

Tom: So to wrap up on "Learning structured linear dynamical systems from missing observations," we have a rigorous framework proving that identification is possible under specific conditions.

Jane: It’s a deep dive into how geometry and probability interact when we try to learn dynamic systems from incomplete records.

Lu: It opens up avenues for understanding complex dynamics where perfect observation is impossible, which is something the AI field needs to tackle more directly.

Meng: For practical AI deployment, this means we have a theoretical floor on what's achievable when data quality degrades over time.

Lalam: And it shows that even with missing information, the structure of the system itself can guide us toward a reliable estimate of A*.

Division of Mathematical Sciences, SPMS, NTU Singapore

stat.ML, cs.LG, cs.SY, eess.SY, math.OC, math.ST, stat.TH

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 62 pages, 3 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 85/100

The gist: As a fastidious and diligent AI researcher, I have meticulously analyzed both provided texts concerning the paper on "Learning structured linear dynamical systems from missing observations." The

Key concepts

Bias-Corrected Objective Function: This is a modified mathematical goal used to find the best estimate for the system matrix $A$. It adjusts a standard least squares problem by accounting for potential errors or biases introduced by missing data, making it more robust than simple least squares.
Projected Gradient Descent (PGD): This is an optimization algorithm used to iteratively search for the optimal matrix $A$ within a specific constraint set $K$. It works by taking steps in the direction that reduces the objective function while ensuring that every step stays within the allowed set of matrices.
Local Complexity ($\gamma_1, \gamma_2$ Functionals): These are mathematical tools used to measure how complex or 'wiggly' the constraint set $K$ is around a specific point. They quantify the structural difficulty of solving the estimation problem at that point, directly influencing how fast an algorithm can converge.
Non-Asymptotic Bounds: These are mathematical guarantees on the error of an estimator that hold for any finite amount of data. Unlike asymptotic bounds (which only work with infinite data), these bounds tell us exactly how large the estimation error will be given a specific trajectory length $T$ and observation probability $p.

Terminology

Objective Function — 5× in this paper · explained in 2 episodes
Sparsity — 4× in this paper · explained in 4 episodes
System Identification — 4× in this paper · explained in 3 episodes
Trace — 3× in this paper · explained in 2 episodes
Convergence Guarantees — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Embeddings — 1× in this paper · explained in 2 episodes
Generative AI — 1× in this paper · explained in 2 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes
IEEE Conference on Decision and Control — 1× in this paper
Non-asymptotic Analysis — 1× in this paper · explained in 2 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes
Reinforcement Learning — 1× in this paper · explained in 4 episodes
Theoretical Guarantees — 1× in this paper · explained in 2 episodes

Summary

As a fastidious and diligent AI researcher, I have meticulously analyzed both provided texts concerning the paper on Learning structured linear dynamical systems from missing observations. The information is highly technical, focusing on non-asymptotic convergence guarantees for estimating a system matrix A* when observations are incomplete.

Here is the comprehensive, detailed summary synthesized from both sources:

Comprehensive Research Summary: Learning Structured Linear Dynamical Systems from Missing Observations

This research paper addresses the challenging problem of finite-time identification of an unknown linear dynamical system matrix A* in R n times n governing the evolution x t+1 = A* x t + eta t+1, where eta t+1 are i.i.d. subgaussian process noise, given only a single trajectory of states (x 0, x 1,, x T) from which some entries are missing with probability p. The core contribution is the development of rigorous non-asymptotic bounds and convergence guarantees for an estimator derived under a bias-corrected least squares objective function constrained to a convex set K.

Problem Formulation and Methodology

The estimation task is framed as minimizing a bias-corrected least squares problem:

A in K (A) = Z - A Z squared F - (1-p) A D squared F q = g(A)

where Z and D are related to the observed data, and F is the Frobenius norm. The analysis hinges on relating the estimation error |A b - A*| F to structural complexity measures of the constraint set K.

The methodology employs Projected Gradient Descent (PGD). Theorem 2 establishes linear convergence of this algorithm (w.h.p.) to A* up to a small additive error epsilon. The learning rate is specifically tuned as gamma = 1/(2p 2T J 2(A*)).

Core Theoretical Results and Error Bounds

The paper establishes two central theorems that provide the non-asymptotic guarantees:

Theorem 1 (Error Bound): This theorem provides a high-probability bound on the estimation error |A b - A*| F. The bound is complex, depending on several structural and statistical parameters:

|A b - A*| F J 3(A*) p squared ((n cubed over 2 3/2 nT, n 1/2 over 2 1/2 n sqrt T) + J 8(A*) n over L 1/3 1/3 n p 5/3T squared/4 + J 5/3(A*)n L 1/3 1/3 n p 5/3T squared/4 )

Crucially, this bound depends on the local complexity of K at A*, quantified by Talagrand’s gamma 1 and gamma 2 functionals. The bound explicitly includes terms involving trajectory length T, sub-sampling probability p, and the distance between a point in the constraint set (B in K) and the true matrix (A* - B).

Theorem 2 (Convergence Rate): This theorem proves that PGD converges linearly (w.h.p.) to A* with a rate rho at most 1 - 1/c J 2(A*). The additive error epsilon in this convergence is bounded by:

epsilon = C J(A*) p squared (n 3/2 over 2 3/2 nT, n 1/2 over 2 1/2 n sqrt T) + n(L n) T 2/3/3 + p (1/delta) sqrt T

Structural Instantiation and Specific Corollaries

The general theory is applied to three specific, structurally rich constraint sets K:

Subspace Case (Corollary 1): For a d-dimensional subspace, the bounds simplify, showing meaningful recovery is possible when T = o(n 2) and p = o(1).
Bi-Isotonic Regression (Corollary 3): When K is the set of bi-isotonic matrices, the error bound depends on k(A*), the minimum number of rectangular blocks for which A* is constant.
Lipschitz Regression: This setting involves matrices whose rows are formed by sampling Lipschitz functions. Remark 17 notes that the projection step in this case can be solved efficiently using interior-point methods, polynomial in n.
Supporting Lemmas and Complexity Analysis

The proofs rely on intricate analysis involving:

Talagrand Functionals (gamma 1, gamma 2): These measure the local complexity of the tangent cone of K at A*, which directly dictate the dependence of the error bounds on K. For instance, gamma 1(CA*, F 2) is related to delta(vec(TK, A*)), which itself is bounded by terms involving p k(A*).
Concentration Inequalities: The analysis utilizes concentration inequalities (Theorems 3-5) to bound the stochastic errors inherent in the PGD updates.
Lemma 16 & 17: These lemmas provide critical bounds on the matrix operator (I - 2 gamma E[M]E) F and its difference (E[M] - M) F, which are essential for establishing the linear convergence rate in Theorem 2.

Conclusion

The paper successfully extends prior work (Tyagi and Efimov, 2024) to the setting of missing values. The core finding is that meaningful recovery of A is possible* under conditions where the trajectory length T scales sub-quadratically with system dimension (T = o(n 2)) and the observation probability p vanishes slowly (p = o(1)). The entire analysis demonstrates that the estimation error is fundamentally controlled by a combination of structural complexity measures (gamma 1, gamma 2) of the constraint set K and statistical factors (T, p). The final results summarize that under appropriate conditions on T, p, and the system's properties (J(A*)), PGD achieves convergence to a small ball around A*.

Improvements for AI systems

Improved estimation of structured linear dynamical systems from missing observations: The system can now provably recover a transition matrix using an estimator that minimizes a bias-corrected least squares problem under a general closed convex set K, even when only a small subset of observations are available at each time point.
Enhanced convergence guarantees for iterative learning algorithms: Projected gradient descent (PGD) now converges linearly to the true system matrix with probability at least 1 − δ, provided the trajectory length T is sufficiently large relative to the complexity measures like local complexity of K and stability parameters.
Recovery under specific structural assumptions: The system can achieve meaningful recovery of the transition matrix in settings where (i) K is a subspace, (ii) K is the set of bi-isotonic matrices, or (iii) K is the set of matrices whose rows are formed by sampling Lipschitz functions, even when T = o(n 2) and p = o(1).
Robustness against missing data imputation strategies: The theoretical framework establishes bounds for estimation based on imputing missing entries by zero, providing provable error rates for structured A∗ recovery in this realistic scenario.
Optimized learning rate selection: The PGD algorithm can utilize a theoretically derived learning rate, specifically γ = 1/2p 2T J 2(A∗), to ensure convergence under the required conditions.

Abstract

We consider the problem of learning structured linear dynamical systems over convex sets K, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of K, the trajectory length T, and the sub-sampling probability p. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i) K is a subspace, (ii) K is the set of bi-isotonic matrices, and (iii) K is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of T much smaller than what is required in the unconstrained case, and for p = o(1).