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arxiv: 2601.22400 · v2 · submitted 2026-01-29 · 🪐 quant-ph · cs.AI

Spectral Filtering for Complex Linear Dynamical Systems

Elad Hazan , Annie Marsden This is my paper

Pith reviewed 2026-05-16 09:09 UTC · model grok-4.3

classification 🪐 quant-ph cs.AI
keywords complex linear dynamical systemsspectral filteringSlepian basisdimension-free regretsector-bounded spectrumonline sequence predictioneffective dimension
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The pith

Learnability of complex linear dynamical systems with sector-bounded spectrum depends only on an effective dimension independent of state space size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that complex-valued linear dynamical systems whose spectrum lies inside a sector of the unit disk can be learned with performance controlled by an effective dimension set only by the sector geometry. A spectral filtering step based on the Slepian basis projects the dynamics onto this low-dimensional subspace, so that online sequence prediction regret stays bounded even as the underlying state dimension grows arbitrarily large. This class includes oscillatory and long-memory behaviors common in signal processing, structured state-space models, and quantum systems. The result therefore supplies a route to scalable prediction algorithms that avoid the usual dependence on ambient dimension.

Core claim

We introduce a spectral filtering method based on the Slepian basis and show that learnability is governed by an effective dimension independent of the ambient state dimension. As a consequence, we obtain dimension-free regret bounds for sequence prediction in CLDS with spectrum contained in a sector of the unit disk.

What carries the argument

Spectral filtering via the Slepian basis, which concentrates the system energy into a low-dimensional subspace whose size is fixed by the spectral sector rather than the full state dimension.

If this is right

  • Regret bounds for online sequence prediction that remain independent of the full state dimension.
  • Efficient learning algorithms for high-dimensional oscillatory and long-memory processes.
  • Practical prediction methods for quantum systems and structured state-space models whose spectra obey the sector restriction.
  • A template for reducing other spectral-constrained linear systems to low effective dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same effective-dimension reduction may apply to related online control or filtering tasks under similar spectral constraints.
  • Concrete numerical tests on high-dimensional quantum or signal examples could quantify the practical gap between effective and ambient dimension.
  • Extending the Slepian projection to other compact spectral sets could enlarge the class of systems admitting dimension-free guarantees.

Load-bearing premise

The spectrum of the complex linear dynamical system lies inside a fixed sector of the unit disk.

What would settle it

A concrete family of CLDS whose spectrum satisfies the sector condition yet whose regret for any sequence predictor grows with the ambient state dimension.

read the original abstract

We study the problem of learning complex-valued linear dynamical systems (CLDS) with sector-bounded spectrum. This class captures oscillatory and long-memory dynamics arising in signal processing, structured state space models, and quantum systems. We introduce a spectral filtering method based on the Slepian basis and show that learnability is governed by an effective dimension independent of the ambient state dimension. As a consequence, we obtain dimension-free regret bounds for sequence prediction in CLDS with spectrum contained in a sector of the unit disk.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies learning complex-valued linear dynamical systems (CLDS) whose spectrum lies in a sector of the unit disk. It introduces a spectral filtering procedure based on the Slepian basis and asserts that learnability is controlled by an effective dimension independent of the ambient state dimension, which in turn yields dimension-free regret bounds for sequence prediction.

Significance. If the central claim is established, the work would supply a dimension-independent learning guarantee for a practically relevant class of oscillatory and long-memory systems arising in signal processing, structured state-space models, and quantum dynamics, thereby extending existing regret analyses beyond the usual dependence on state dimension.

major comments (1)
  1. Abstract: the central claim that learnability is governed by an effective dimension independent of ambient state dimension is stated without any theorem statement, derivation, error bound, or proof sketch; the available text supplies no technical support for the dimension-free regret result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [—] Abstract: the central claim that learnability is governed by an effective dimension independent of ambient state dimension is stated without any theorem statement, derivation, error bound, or proof sketch; the available text supplies no technical support for the dimension-free regret result.

    Authors: The abstract is a concise summary of the paper's contributions and is not intended to contain theorem statements, derivations, error bounds, or proof sketches. The full manuscript supplies these technical details, establishing that the effective dimension depends only on the sector parameters and Slepian basis properties (independent of state dimension) and deriving the corresponding dimension-free regret bounds for sequence prediction. revision: no

Circularity Check

0 steps flagged

No circularity detected from abstract

full rationale

Only the abstract is available, which states the problem setup for CLDS with sector-bounded spectrum, introduces a spectral filtering method based on the Slepian basis, and claims that learnability is governed by an effective dimension independent of ambient state dimension, yielding dimension-free regret bounds. No equations, parameters, fitted quantities, or self-citations appear. The spectral restriction is presented as an explicit assumption required for the bounds, not derived from prior steps within the paper. No load-bearing derivation chain is visible that reduces to its own inputs by construction, so the analysis finds no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the explicit assumptions stated there.

axioms (1)
  • domain assumption The system is a complex-valued linear dynamical system whose spectrum lies inside a sector of the unit disk.
    This spectral restriction is the defining property of the class studied and is required for the effective-dimension claim.

pith-pipeline@v0.9.0 · 5336 in / 1121 out tokens · 20255 ms · 2026-05-16T09:09:45.661427+00:00 · methodology

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