arxiv: 2604.08913 · v1 · submitted 2026-04-10 · 🧮 math.OC · cs.SY· eess.SY

Recognition: no theorem link

Connections Between Determinantal Point Processes and Gramians in Control

Ahmad F. Taha, Mohamad H. Kazma

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:51 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords determinantal point processesobservability Gramiancontrollability Gramiansensor selectionactuator selectionlinear dynamic systemsnegative dependencediversity

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The pith

Observability and controllability Gramians act as determinantal point processes when parameterized over sensor and actuator node subsets.

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

The paper shows that the Gramian matrix, when restricted to subsets of sensors or actuators, defines the kernel of a determinantal point process. This assigns higher probability to node collections whose information content complements rather than overlaps. A reader cares because the link supplies a probabilistic and spectral account of why certain selections outperform others in linear systems, without invoking stochastic dynamics. It yields an effective rank condition, monotonicity, negative dependence, and a maximum-a-posteriori reading of the classic selection task while recovering greedy guarantees. The result reframes sensor placement as sampling from a diversity-favoring measure induced by the Gramian.

Core claim

By showing that the observability (controllability) Gramian parameterized by sensor (control) node subsets is a DPP, the work supplies a probabilistic and spectral perspective on sensor (actuator) selection for linear dynamic systems. The induced probability measure over configurations favors diverse selections, produces an effective observable rank condition, balances individual node contributions against collective diversity, and satisfies node-inclusion monotonicity together with negative dependence. The same formulation recovers classical greedy optimization bounds and admits a maximum-a-posteriori interpretation of the placement problem.

What carries the argument

The Gramian matrix parameterized by node subsets, serving as the kernel matrix that defines subset probabilities via its principal minors.

If this is right

Sensor and actuator selection acquires an explicit probabilistic interpretation in which diverse configurations receive higher likelihood.
The induced measure exhibits negative dependence, so adding one informative node lowers the marginal probability of redundant others.
Node-inclusion monotonicity holds: enlarging a feasible set never decreases the probability of its subsets.
Classical greedy algorithms for Gramian-based optimization are recovered as approximations to the maximum-a-posteriori solution.
An effective rank condition on the Gramian replaces the classical full-rank observability test for subset selection.

Where Pith is reading between the lines

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

DPP sampling algorithms could be imported directly to generate near-optimal sensor sets without exhaustive search.
The same parameterization might apply to other quadratic forms arising in control, such as solutions of Lyapunov or Riccati equations.
In large networks the spectral view could guide placement heuristics that explicitly penalize information overlap rather than just trace or determinant objectives.

Load-bearing premise

The Gramian submatrices, when indexed by every possible node subset, must induce a valid probability measure obeying the negative-dependence and diversity properties of a determinantal point process.

What would settle it

A concrete linear system and choice of nodes where the determinant-based probabilities either fail to sum to one across all subsets or assign higher likelihood to redundant rather than complementary selections.

Figures

Figures reproduced from arXiv: 2604.08913 by Ahmad F. Taha, Mohamad H. Kazma.

**Figure 2.** Figure 2: Geometric view of the Gramian DPP equivalence ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: (a) Effective observable rank validation, theoretical value neff = trace(K) is equivalent to the empirical mean cardinality E[|Y |] over Mval = 200 DPP samples. (b) Observability mode activation probabilities κi = λi/(λi + 1). Bars and curves show means over 20 realizations; bands indicate ±1 standard deviation. To address the above questions, we consider three LTI network models of increasing complexity… view at source ↗

**Figure 6.** Figure 6: For M = 500 k-DPP samples at r = 10 and for all 20 realizations. Each point represents one sample. The x-axis is the quality term P i∈Y log q 2 i and the y-axis is the diversity term logdet(SY ). Furthermore, to validate inclusion monotonicity, we vary the stability margin α ∈ {0.5, 0.7, 0.85, 0.95}, scaling A ← αA/ρ(A), such that ρ(A) = α < 1. For α1 < α2 this implies that W(α1) o ⪯ W(α2) o . According t… view at source ↗

read the original abstract

Determinantal point processes (DPPs) are probability models over subsets of a ground set that favor diverse selections while suppressing redundancy. That is, they tend to assign higher likelihood to collections whose elements complement one another instead of repeating the same information. For example, in recommendation systems, a DPP prefers showing users several relevant items that differ in content or style, rather than many near-duplicates of essentially the same item. Although DPPs have been studied extensively in machine learning, random matrix theory, and popularized through components of YouTube's search recommendation system, they have not been considered in the context of dynamic systems; time domain analysis is not a feature of DPPs. This paper establishes interesting connections between DPPs and control theory. By showing that the observability (controllability) Gramian parameterized by sensor (control) node subsets is a DPP, we provide a probabilistic and spectral perspective on sensor (actuator) selection for linear dynamic systems. This notion of probability here does not represent stochastic uncertainty in the system dynamics; it instead represents a likelihood measure over sensor (actuator) configurations induced by the Gramian. To that end, we derive an effective observable rank condition, characterize the balance between individual node contributions and diversity, and establish node inclusion monotonicity and negative dependence properties. Finally, we show that this formulation recovers classical greedy optimization guarantees and admits a maximum a posteriori interpretation of the sensor/actuator node selection problem. Numerical case studies on three network topologies corroborate the theoretical results.

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.

Desk Editor's Note private letter to a colleague

The claimed link between Gramians and DPPs does not hold, since det of summed per-node Gramians is not a principal minor of any fixed kernel.

read the letter

The paper's central claim is that the observability or controllability Gramian parameterized over sensor or actuator subsets forms a determinantal point process. This does not work. DPPs require the subset probability to be proportional to the determinant of a principal submatrix drawn from one fixed kernel matrix K. The subset Gramian is instead an additive sum of individual node Gramians, and the determinant of that sum does not arise as det(K_S) for any single K in general. The two-node counterexample with orthogonal diagonal contributions already produces det values that no real kernel can match: singles are zero while the pair is positive. That mismatch is load-bearing for the advertised probabilistic lens. The abstract and title both rest on this identification, so the connection to DPPs is not established. What the paper does establish are several properties of the measure induced by det of the summed Gramian. It gives an effective rank condition, discusses the balance between single-node strength and diversity, proves node-inclusion monotonicity, shows negative dependence for this particular measure, recovers the standard greedy approximation bounds, and gives a MAP reading of the selection problem. These steps appear to be carried out directly on the Gramian and do not rely on the DPP step, so they may be valid on their own. The numerical examples on three network topologies are used to check the selection behavior in practice. The main soft spot is therefore the overstated DPP framing rather than any obvious error in the derived properties themselves. Readers who want a genuine bridge to DPPs from machine learning will be disappointed, but the underlying selection criteria and their monotonicity and submodularity features could still be useful for network control work if the paper is reframed. This is aimed at control theorists who handle sensor and actuator placement in linear systems and who are open to combinatorial or probabilistic views of the problem. It shows honest engagement with both the control and DPP literatures even though the analogy does not land. I would send it to peer review so that referees can check the derivations of the properties and decide whether the contribution survives once the DPP claim is removed or corrected.

Referee Report

3 major / 2 minor

Summary. The paper claims that parameterizing the observability (controllability) Gramian over subsets of sensor (actuator) nodes induces a determinantal point process (DPP). This supplies a probabilistic/spectral view of node selection in linear systems, from which the authors derive an effective rank condition, node-contribution/diversity balance, inclusion monotonicity, negative dependence, recovery of greedy submodular guarantees, and a MAP interpretation of the selection problem. The probability measure is defined via the Gramian determinant rather than stochastic dynamics.

Significance. A correct link between additive Gramian determinants and DPP kernels would give control theorists access to DPP tools for explaining diversity in sensor placement and for analyzing greedy algorithms. The manuscript supplies no machine-checked proofs, reproducible code, or parameter-free derivations that would strengthen the result if the identification held.

major comments (3)

[Abstract] Abstract and central claim: the assertion that 'the observability (controllability) Gramian parameterized by sensor (control) node subsets is a DPP' does not hold. DPPs require a probability measure of the form P(S) ∝ det(K_S) for a single fixed kernel matrix K on the ground set. The paper instead uses det(W_S) where W_S = ∑_{i∈S} W_i and each W_i is a per-node positive-semidefinite Gramian. These two forms are not equivalent in general.
[Abstract] Abstract / §3 (presumed derivation of the DPP property): the counter-example with a two-state system, A = 0, finite horizon T, W_1 = T·diag(1,0), W_2 = T·diag(0,1) yields det(W_{{1}}) = det(W_{{2}}) = 0 yet det(W_{{1,2}}) = T² > 0. No real matrix K satisfies K_{11} = K_{22} = 0 and det(K_{{1,2}}) = −K_{12}² = T². Hence the induced measure cannot be a DPP, and all subsequent appeals to DPP negative dependence or diversity properties rest on an invalid identification.
[§4–5] §4–5 (monotonicity, negative dependence, greedy recovery): these properties are derived under the DPP label. Because the measure is not a DPP, the derivations must be re-checked directly from the additive Gramian structure; the manuscript provides no such independent verification.

minor comments (2)

Clarify whether the authors intend a different kernel construction or a restricted class of Gramians (e.g., rank-1) under which det(W_S) coincides with a DPP.
[Abstract] The abstract states that 'time domain analysis is not a feature of DPPs'; this observation is correct but should be expanded to explain why the finite-horizon Gramian still yields a valid probability measure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and for pointing out the inaccuracy in our identification of the Gramian-based measure with a determinantal point process. We agree that this claim does not hold in general and will revise the manuscript to address this issue.

read point-by-point responses

Referee: [Abstract] Abstract and central claim: the assertion that 'the observability (controllability) Gramian parameterized by sensor (control) node subsets is a DPP' does not hold. DPPs require a probability measure of the form P(S) ∝ det(K_S) for a single fixed kernel matrix K on the ground set. The paper instead uses det(W_S) where W_S = ∑_{i∈S} W_i and each W_i is a per-node positive-semidefinite Gramian. These two forms are not equivalent in general.

Authors: We acknowledge that the referee is correct. The standard DPP definition relies on the determinant of a principal submatrix of a fixed kernel matrix K. Our construction uses the determinant of a sum of matrices, which is distinct. This was an error in the manuscript's central claim. We will revise the abstract and introduction to remove the assertion that the Gramian induces a DPP and instead present it as a determinantal measure arising from additive Gramians. revision: yes
Referee: [Abstract] Abstract / §3 (presumed derivation of the DPP property): the counter-example with a two-state system, A = 0, finite horizon T, W_1 = T·diag(1,0), W_2 = T·diag(0,1) yields det(W_{{1}}) = det(W_{{2}}) = 0 yet det(W_{{1,2}}) = T² > 0. No real matrix K satisfies K_{11} = K_{22} = 0 and det(K_{{1,2}}) = −K_{12}² = T². Hence the induced measure cannot be a DPP, and all subsequent appeals to DPP negative dependence or diversity properties rest on an invalid identification.

Authors: The provided counterexample is valid and illustrates the fundamental difference between the two constructions. No such kernel K exists for this case, confirming that the measure is not a DPP. We will incorporate this example into the revised manuscript to clarify the distinction and will not rely on DPP-specific properties without independent justification. revision: yes
Referee: [§4–5] §4–5 (monotonicity, negative dependence, greedy recovery): these properties are derived under the DPP label. Because the measure is not a DPP, the derivations must be re-checked directly from the additive Gramian structure; the manuscript provides no such independent verification.

Authors: We agree that the properties should be verified independently of the DPP label. Inclusion monotonicity follows directly from the fact that adding a positive semidefinite matrix cannot decrease the determinant. For negative dependence and the recovery of greedy guarantees, we will provide direct proofs based on the properties of the Gramian determinant in the revision. If some properties do not hold in the same form, we will note the differences. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation presents Gramian parameterization as inducing DPP-like measure without definitional reduction or load-bearing self-citation

full rationale

The paper's central step defines a measure over sensor subsets via det(W_S) where W_S is the additive Gramian sum, then derives properties (negative dependence, monotonicity, greedy guarantees) directly from the Gramian structure and linear systems theory. This is not equivalent to its inputs by construction, nor does it rely on fitting parameters to the target or importing uniqueness via self-citation chains. External DPP literature is referenced for context but not used to justify the identification itself. The derivation chain remains self-contained against the stated assumptions on the Gramian.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard linear systems theory and the definition of Gramians; no free parameters or new entities are indicated in the abstract.

axioms (2)

domain assumption The underlying system is linear time-invariant
Required to define the controllability and observability Gramians in the standard way.
domain assumption The Gramian can be meaningfully parameterized by subsets of nodes
This parameterization is the step that allows the DPP connection to be made.

pith-pipeline@v0.9.0 · 5575 in / 1359 out tokens · 45529 ms · 2026-05-10T17:51:54.631769+00:00 · methodology

discussion (0)

Reference graph

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