Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions

Mohit Kumar; Somayeh Kargaran

cs.AISep 26, 2026

Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions

Authors: Mohit Kumar, Somayeh Kargaran

Organizations: Software Competence Center Hagenberg, Hagenberg, Upper Austria, Austria

Abstract

A soft state representation assigns each state a vector of nonnegative class weights that sum to one. We study how the construction of these weights and the state dynamics jointly determine the accuracy of linear prediction. For any fixed measurable representation, we derive a finite-sample lower confidence bound on the smallest population root-mean-square prediction error among matrices with a specified spectral-norm limit. The bound compares variation in successor coordinates within each reference class with the improvement that soft inputs could provide. It is computed from independent evaluation pairs without fitting a prediction matrix. A bound above a chosen tolerance rules out that tolerance for the entire matrix class; a zero bound is inconclusive. For coordinates constructed using Kernel Affine Hull Machines, reconstruction-score margins control disagreement with reference labels and enter bounds on prediction error. Under exact deterministic linear evolution, we also establish the Koopman and reproducing-kernel Hilbert-space adjoint interpretation, accounting for redundant coefficient vectors. A four-state study compares the confidence bound with analytically known optima across 117,000 reported replicate datasets. A Van der Pol representation selected on pilot data is then evaluated on 32 independent datasets under each of two transition laws. The reported bounds are positive at the fitted matrix norm, but can become zero at larger norm limits. Further forecasting studies examine coordinate variation, common prediction targets, and long-horizon error. The results distinguish agreement with reconstruction classes, attainable prediction accuracy, and exact operator closure.

Figures & tables

Question	Supporting result	Conditions and interpretation
Can any permitted linear predictor meet a chosen error tolerance?	Population lower bound and finite-sample certificate.	Fixed representation and evaluation law; a bound above the tolerance excludes it for the norm-bounded class. A zero bound is inconclusive.
How does reconstruction quality affect prediction?	Bounds based on reconstruction-score margins and a four-state example.	The margin bounds control agreement with labels. Successor variation must also be considered.
When is coordinate prediction an exact Koopman representation?	Finite-feature RKHS and reduced spectral correspondence.	Deterministic dynamics, exact closure on the domain, and nonzero represented functions.
How well do fitted models forecast the chosen coordinates?	Forecasting experiments on specified targets.	Conclusions depend on the target, horizon, selection procedure, and sampling unit.

Table 1: Questions addressed by the manuscript and the conditions needed to interpret the answers.

Symbol	Meaning
$\phi$ , $z$ , $y$	Current soft coordinates, successor soft coordinates, and the current one-hot reference label.
$\mathcal{D}_{\kappa}(\Phi)$	Minimum population root-mean-square error over matrices with $\\|B\\|_{2}\leq\kappa$ .
$\sigma_{\mathrm{res}}$	Root-mean-square variation of successor coordinates about their class mean.
$a_{\Phi}$	Root-mean-square Euclidean distance between the soft coordinates and the one-hot label.
$A_{\mathrm{fid}}$	Mean squared shortfall of the weight assigned to the reference class; $a_{\Phi}^{2}\leq 2A_{\mathrm{fid}}$ .
$\widehat{f}$ , $\widehat{s}$	Evaluation-sample assignment loss and within-class successor variance.

Table 2: Quantities used in the prediction-error bound. Every population quantity refers to the same fixed representation and evaluation law.

	Bound from uniform control of risk	Bound based on reference labels
Empirical quantity	Minimum sample prediction error over the matrix class, or a certified lower bound on it.	Assignment loss and within-class successor variance.
Role of labels	Uses coordinate inputs and successor targets directly.	Groups successor vectors by their current reference label.
Use of soft coordinates	Finds their best linear predictor within the matrix class.	Bounds the possible reduction in error relative to hard-label prediction.
Main limitation	Needs a uniform confidence correction and a valid lower bound on minimum empirical risk.	May be zero despite positive minimum error, especially with few samples per class or large deviations from one-hot labels.

Table 3: Two lower bounds on the minimum prediction error for fixed coordinates. Numerical comparisons require both confidence guarantees to cover the same evaluation distribution and prediction-matrix class.

Fig. 1: Exact minimum prediction error (solid curve) and limiting lower confidence bound (dashed curve) for the four-state family in ( 18 ), with

\kappa=\sqrt{2}

. The curves are evaluated directly from the analytical formulas. The lower bound is zero for

p\leq 5/6

, although the optimum is positive for

p>1/2

. Their difference can persist with unlimited evaluation data

System	Evaluation law	$\\|\widehat{B}\\|_{2}$	$\overline{\widehat{f}}$	$\overline{\widehat{s}}$	Critical norm
Duffing	Matched stochastic	1.005768	0.780756	0.000217	0.011796
Duffing	Deterministic	1.005768	0.780787	0.000222	0.011914
Van der Pol	Matched stochastic	1.078133	0.360983	0.017542	0.155876
Van der Pol	Deterministic	1.078133	0.361292	0.017328	0.154858

Table 4: Pilot evaluation of the original representations at

M=2048

, with eight replicate datasets per row. Bars denote means across replicates. The final column gives the planning value

\sqrt{\overline{\widehat{s}}/(2\overline{\widehat{f}})}

. The displayed inputs are rounded and may not reproduce the final digits of that column.

Evaluation law	Positive $L$	Median $L$ [5th, 95th]	Minimum $L$	Median RMSE
Matched stochastic	32/32	0.084727 [0.078444, 0.089910]	0.076333	0.276233
Deterministic	32/32	0.084269 [0.077720, 0.092126]	0.075401	0.275797

Table 5: Van der Pol evaluation using independent pairs at the fitted norm

\kappa=1.0203452494452896

, with

\delta=0.05

and

M=16{,}384

pairs per replicate. Each law has 32 independent replicate datasets. Brackets give the descriptive 5th and 95th percentiles across replicates, rather than confidence limits. Empirical RMSE is defined in ( 17 ).

	Fitted matrix	Positive $L$		Median $L$
Norm limit	permitted?	Stochastic	Deterministic	Stochastic	Deterministic
$1$	No	32/32	32/32	0.089186	0.088660
$\\|\widehat{B}\\|_{2}$	Yes	32/32	32/32	0.084727	0.084269
$\sqrt{2}$	Yes	11/32	8/32	0	0
$2\\|\widehat{B}\\|_{2}$	Yes	0/32	0/32	0	0

Table 6: All four prespecified norm limits in the Van der Pol study. Counts give positive certificates among 32 replicates per law. The fitted-matrix column applies under both laws:

\widehat{B}

lies outside the class with

\kappa=1

and inside the other three classes.

Dynamics	$\sigma_{\mathrm{res}}$	Optimal RMSE	Certificate limit
Identity: $F(x)=x$	$0$	$0$	$0$
Conflicting successors: $F(a)=a$ ; $F(b)=F(d)=F(e)=d$	$15/32$	$15/32$	$13/32$

Table 7: One fixed KAHM representation, two dynamics maps. At

p=31/32

, both cases have assignment loss

1/1024

and coordinate variance

225/512

. The norm limit is

\sqrt{2}

. The final column is the certificate limit as the number of i.i.d. evaluation pairs grows.

$C$	$\omega$	Mean $R^{2}$	Centered error	$\rho_{\mathrm{var}}$	$r_{\mathrm{eff}}$	$\rho_{\mathrm{rank}}$	Selected
25	6	0.8885	$0.1115\pm 0.0034$	0.3545	14.26	0.5942	Yes
30	6	0.8865	$0.1135\pm 0.0030$	0.3219	15.58	0.5373	No
20	4	0.8865	$0.1135\pm 0.0063$	0.2720	10.42	0.5482	No

Table 8: The complete one-standard-error set for Van der Pol selection without added observation noise. Centered error is the per-run mean of

1-R_{h}^{2}

over five horizons, summarized as mean

\pm

standard error across three fitted runs. Variation diagnostics are averaged across runs. All values were used in model selection.

	Duffing, $C=15$ , $\omega=12$			Van der Pol, $C=20$ , $\omega=4$
Coordinate map	$E_{1}$	$R_{1}^{2}$	$E_{200}$	$E_{1}$	$R_{1}^{2}$	$E_{200}$
KAHM affinities	$5.74\!\times\!10^{-6}$	0.9942	$5.44\!\times\!10^{-4}$	$1.89\!\times\!10^{-3}$	0.9976	0.180
$K$ -means RBF	$6.87\!\times\!10^{-5}$	0.9950	$2.45\!\times\!10^{-2}$	$3.01\!\times\!10^{-2}$	0.9672	0.819
$K$ -means distance	$1.92\!\times\!10^{-3}$	0.9924	0.583	$9.21\!\times\!10^{-2}$	0.9001	1.07
$K$ -means hard	$2.12\!\times\!10^{-2}$	0.9411	0.393	0.112	0.8792	1.09

Table 9: Comparison of coordinate constructions at fixed oscillator configurations. Entries are means over three runs; sample standard deviations are supplied in the released per-run summary files. Each map is evaluated in its own coordinate space using the same NLMS estimator. Thus the table measures the predictability of each map’s own coordinates.

Training trajectories	Across noise levels $R^{2}$	No added noise $R^{2}$	Reference $R^{2}$	$\Delta J_{\mathrm{clean-noise}}$	$\Delta J_{\mathrm{ref-noise}}$
1	$0.8745\pm 0.0055$	$0.8768\pm 0.0134$	$0.8795\pm 0.0081$	$-0.0022\pm 0.0055$	$-0.0049\pm 0.0077$
2	$0.8876\pm 0.0050$	$0.8840\pm 0.0047$	$0.8889\pm 0.0125$	$0.0036\pm 0.0030$	$-0.0013\pm 0.0064$
3	$0.8930\pm 0.0062$	$0.8936\pm 0.0065$	$0.8982\pm 0.0044$	$-0.0005\pm 0.0033$	$-0.0052\pm 0.0051$
5	$0.8985\pm 0.0029$	$0.8861\pm 0.0027$	$0.8958\pm 0.0070$	$0.0124\pm 0.0024$	$0.0028\pm 0.0036$
8	$0.9021\pm 0.0100$	$0.8891\pm 0.0045$	$0.9077\pm 0.0030$	$0.0130\pm 0.0032$	$-0.0056\pm 0.0070$

Table 10: Van der Pol prediction as the number of training trajectories increases. The configurations selected across observation-noise levels, without added observation noise, and used as a reference are

(C,\omega)=(25,4)

(25,6)

, and

(20,4)

, respectively. The

R^{2}

columns give means over five horizons, summarized as mean

\pm

standard deviation across three training-seed blocks. The last two columns give paired differences in centered error

J

, as mean

\pm

standard error. These are descriptive comparisons of representation-specific targets.

System	Method	$R_{1}^{2}$	$R_{50}^{2}$	$R_{200}^{2}$
Duffing	Coordinate NLMS	$0.9835\pm 0.0003$	$0.5056\pm 0.0167$	$-0.3155\pm 0.0552$
	State DMD	$0.9895\pm 0.0021$	$-0.7108\pm 0.1433$	$-15.0419\pm 1.7772$
	State EDMD, degree 2	$0.9984\pm 0.0015$	$0.9975\pm 0.0019$	$0.9546\pm 0.0134$
	State EDMD, degree 3	$0.9990\pm 0.0017$	$0.9961\pm 0.0009$	$0.9774\pm 0.0114$
Van der Pol	Coordinate NLMS	$0.9963\pm 0.0002$	$0.9020\pm 0.0024$	$0.7724\pm 0.0044$
	State DMD	$0.9574\pm 0.0015$	$0.0909\pm 0.0037$	$0.2051\pm 0.0019$

Table 11: Oscillator prediction in a fixed KAHM output representation. Within each system all methods share the same target:

(C,\omega)=(10,2)

for Duffing and

(25,4)

for Van der Pol. Entries are mean

R_{h}^{2}\pm

standard deviation over three evaluation trajectories for one fitted model per method. The DMD/EDMD polynomial degrees and ridge regularization are prespecified.

Task	$(C,\omega)$	$E_{1}$	$R_{1}^{2}$	$E_{50}$	$R_{50}^{2}$
CartPole	$(20,1)$	$0.0716\pm 0.0061$	$-0.133\pm 0.189$	$0.0768\pm 0.0115$	$-0.235\pm 0.146$
MountainCar	$(100,1)$	$0.00386\pm 0.00058$	$0.937\pm 0.006$	$0.1305\pm 0.0047$	$-1.111\pm 0.079$
Acrobot	$(150,0.5)$	$0.0096\pm 0.0014$	$0.7566\pm 0.0307$	$0.0577\pm 0.0019$	$-0.3860\pm 0.2369$

Table 12: Exploratory coordinate-prediction diagnostics at the reported configurations under fixed randomized behavior policies. Entries are mean

\pm

standard deviation over three fitted runs with base seeds

0,1,2

; Acrobot standard deviations use divisor

3

. The

R^{2}

scores compare prediction error with always predicting the mean evaluation-target vector. Overlapping episode reset seeds introduce dependence across the base-seed runs.

Predictor	$E_{1}$	$R_{1}^{2}$	$E_{50}$	$R_{50}^{2}$
Direct coordinates, NLMS	$0.0096\pm 0.0014$	$0.7566\pm 0.0307$	$0.0577\pm 0.0019$	$-0.3860\pm 0.2369$
Affine state predictor	$0.0096\pm 0.0003$	$0.7535\pm 0.0325$	$0.0663\pm 0.0150$	$-0.5856\pm 0.3776$
Quadratic state predictor	$0.0059\pm 0.0001$	$0.8475\pm 0.0208$	$0.0806\pm 0.0163$	$-0.8703\pm 0.0928$
Cubic state predictor	$0.0030\pm 0.0003$	$0.9247\pm 0.0076$	$0.0790\pm 0.0083$	$-0.9422\pm 0.5819$

Table 13: Acrobot predictors evaluated against the same KAHM target,

C=150

and

\omega=0.5

. Entries are means

\pm

standard deviations over three fitted runs; the stored standard deviations use divisor

3

. Polynomial state predictions use the specified per-step clipping rule.

Fig. 2: A descriptive view of the MountainCar abstraction at

C=100

\omega=1

, using training seed

0

and evaluation seed

2000

. Colors identify the largest soft coordinate along visited position-velocity states. The 100 coordinate indices are nominal labels, so color differences carry no quantitative meaning. The view illustrates the visited trajectories and coordinate assignments; it does not assess class separation or predictive sufficiency

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Study family	Role here	Selection rationale
Four-state certificate evaluation (21)	Checks against an analytical optimum.	The known population optimum allows coverage checks. Counts and tolerance decisions are reported separately from the dynamical pilot.
Dynamical pilot and subsequent certificate evaluation (22-23)	Application to one selected representation.	Zero pilot outcomes, evaluation on new pairs after selection, and results at all four norm limits are included.
Coordinate construction (6-7) and selection by centered error (9, 12)	Representation diagnostics and hyperparameter selection.	Each map predicts its own coordinates; variation diagnostics help interpret these changing targets.
Increasing training-trajectory count (13)	Effect of additional training data.	All five training counts and three configurations are included. Each map is reconstructed and predicts its own coordinates.
State-space predictor baselines (15)	Comparison on a common target.	Direct coordinate and state-space predictors share an output map within each oscillator system. Both systems and all reported methods are included.
Classic Control and Acrobot (16, 18)	Main fixed-policy diagnostics.	One-step and 50-step scores are both retained, including negative long-horizon outcomes. The measured outcome is coordinate prediction under fixed policies.

Appendix

Table 14: Purpose and interpretation of the reported studies. Experiment numbers are the source identifiers recorded in the manuscript.

Study	$C$	$\omega$	Count	Objective
Duffing reference	$15,\allowbreak 25,\allowbreak 35$	$4,\allowbreak 8,\allowbreak 12$	9	$E_{1}$
Duffing expanded	$8,\allowbreak 10,\allowbreak 12,\allowbreak 15,\allowbreak 20,\allowbreak 25,\allowbreak 30,\allowbreak 40$	$2,\allowbreak 4,\allowbreak 6,\allowbreak 8,\allowbreak 10,\allowbreak 12,\allowbreak 16,\allowbreak 20$	64	$\overline{E}_{\mathcal{H}}$
Van der Pol, no added observation noise	$10,\allowbreak 15,\allowbreak 20,\allowbreak 25,\allowbreak 30,\allowbreak 40,\allowbreak 50$	$0.5,\allowbreak 1,\allowbreak 2,\allowbreak 4,\allowbreak 6,\allowbreak 8,\allowbreak 12$	49	$\overline{(1-R^{2})}_{\mathcal{H}}$ ; 1-SE
Van der Pol, across observation-noise levels	$10,\allowbreak 15,\allowbreak 20,\allowbreak 25,\allowbreak 30,\allowbreak 40$	$0.25,\allowbreak 0.5,\allowbreak 1,\allowbreak 2,\allowbreak 4$	30	weighted $(1-R^{2})$ ; 1-SE
Acrobot	$20,\allowbreak 40,\allowbreak 50,\allowbreak 75,\allowbreak 100,\allowbreak 150$	$0.5,\allowbreak 1,\allowbreak 2,\allowbreak 4,\allowbreak 8$	30	$E_{50}$

Appendix

Table 15: Reported candidate grids. Count is the number of

(C,\omega)

combinations and excludes repeated fits. Duffing searches and Van der Pol selection without added observation noise use seeds

0,1,2

; Van der Pol selection across noise levels additionally uses five training-noise levels. Acrobot selection uses seed

0

, followed by assessment of the selected setting with seeds

0,1,2

. Here

\mathcal{H}=\{1,10,50,100,200\}

. “1-SE” means the one-standard-error rule, with normalized variance and then normalized effective rank used to resolve remaining choices. The weighted centered-error score uses horizons

50,100,200

with weights

1/4,1/4,1/2

Operation	Git revision prefix
Fix confirmatory protocol and configuration	18c2bb4
Build training-only $\omega=128$ representation and matrix	4a0cedd
Recompute the representation and matrix for verification	cbfe438
Generate 64 independent-pair evaluation datasets	18dc9f3
Recompute results from the completed evaluation	090eac5
Generate summaries and repeat numerical verification	689d40d

Appendix

Table 16: Source revisions recorded for the dynamical certificate evaluation. Prefixes identify the full commits listed by the manuscript’s provenance metadata; they are not independent confirmations of execution.

System	Rollout rule	$E_{1}$	$E_{20}$	$E_{200}$
Duffing	No projection	$2.09\times 10^{-5}$	$5.11\times 10^{-4}$	$9.14\times 10^{-4}$
	Per-step vector projection	$2.08\times 10^{-5}$	$5.09\times 10^{-4}$	$1.07\times 10^{-3}$
	Row-stochastic matrix	$4.40\times 10^{-5}$	$7.71\times 10^{-3}$	$1.59\times 10^{-1}$
Van der Pol	No projection	$1.89\times 10^{-3}$	$2.60\times 10^{-2}$	$1.80\times 10^{-1}$
	Per-step vector projection	$1.84\times 10^{-3}$	$2.22\times 10^{-2}$	$1.53\times 10^{-1}$
	Row-stochastic matrix	$3.93\times 10^{-3}$	$3.14\times 10^{-1}$	$9.56\times 10^{-1}$

Appendix

Table 17: Projection diagnostics at the reference configurations. Each system uses a fixed data trajectory and three KAHM-derived abstraction initializations; entries are mean relative squared errors. These runs differ from those in Table 9 . The corresponding mean Duffing one-step

R^{2}

0.9861

System	Estimator	$E_{1}$	$R_{1}^{2}$	$\widehat{r}_{\Phi}/C$	$\operatorname{spr}(M_{\mathrm{red}})$	Min. $\rho$	Median $\rho$	Max. $\eta_{\mathrm{rep}}$	Counts
Duffing	NLMS	$2.50\times 10^{-4}$	0.9835	$10/10$	1.000	$2.75\times 10^{-5}$	$5.88\times 10^{-2}$	$4.58\times 10^{-10}$	$3/4/9$
	Ridge least squares	$2.41\times 10^{-4}$	0.9842	$10/10$	1.000	$9.52\times 10^{-7}$	$6.44\times 10^{-4}$	$3.66\times 10^{-12}$	$9/10/10$
Van der Pol	NLMS	$3.05\times 10^{-3}$	0.9963	$25/25$	1.000	$1.89\times 10^{-5}$	$1.75\times 10^{-4}$	$1.32\times 10^{-14}$	$23/25/25$
	Ridge least squares	$3.03\times 10^{-3}$	0.9964	$25/25$	1.000	$1.91\times 10^{-5}$	$1.01\times 10^{-4}$	$1.13\times 10^{-14}$	$25/25/25$

Appendix

Table 18: Aggregate spectral diagnostics on held-out data. Candidates are eigenpairs of

M_{\mathrm{red}}^{\top}

, the transpose of the fitted prediction matrix expressed on the retained training-feature subspace. The squared residual ratio

\rho

and coefficient-representation error

\eta_{\mathrm{rep}}

use ( 223 ) on a fixed 800-pair held-out subsample. The count triple gives the numbers of valid candidates satisfying

\rho\leq 0.01

0.05

, and

0.10

, respectively. Here

\widehat{r}_{\Phi}=C

;

\operatorname{spr}

denotes spectral radius.

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