Extending SSMs with the Exponentially Weighted Signature

Alexandre Bloch; Benjamin Walker; Joël Mouterde; Sam Morley; Samuel N. Cohen; Terry Lyons

stat.MLMar 19, 2026

Extending SSMs with the Exponentially Weighted Signature

Authors: Alexandre Bloch, Benjamin Walker, Joël Mouterde, Sam Morley, Samuel N. Cohen, Terry Lyons

Organizations: Mathematical Institute, University of Oxford · SKF · Department of Mathematics, Imperial College London

Abstract

We introduce the exponentially weighted signature (EWS), a continuous-time model that computes iterated integrals of a path, where each increment is weighted by the matrix exponential of a learnable generator over elapsed clock time. We prove that it solves a linear controlled differential equation, keeps the group-like structure and the universality of the signature, and satisfies a modified Chen identity, enabling a parallel scan. At depth one the EWS is a state-space model (SSM), and we map linear time-invariant SSMs, Mamba channels and Mamba- $2$ heads to it in closed form. The EWS extends SSMs through an arbitrary matrix generator, a clock that generalises the step size to causal functionals of the input, and higher truncation depths that are non-linear in the path within a single layer. Empirically, the EWS achieves the highest average accuracy and rank on six long time-series classification datasets, where depth generally helps. Learned clocks prove necessary for state tracking on formal language tasks, and at depth one, the EWS matches or exceeds competing SSMs on regression and forecasting with far fewer parameters.

Figures & tables

Family	Spectrum	Kernels
Zero	$\{0\}$	$1$
Diagonal	real	$e^{-\lambda r}$
Rotation	$a\pm i\omega$	$e^{-ar}\cos(\omega r),\ e^{-ar}\sin(\omega r)$
Dense	general	mixtures
Jordan, real	$\lambda$ , repeated	$r^{\ell}e^{-\lambda r}$
Jordan, complex	$a\pm i\omega$ , repeated	$r^{\ell}e^{-ar}\cos(\omega r),\ r^{\ell}e^{-ar}\sin(\omega r)$

Table 1: Some generator families and clocks. A clock is: invariant (i) when the layer does not depend on the path’s parametrisation; stopping (s) when

f

can vanish on non-zero increments.

	Signature		EFM		EWS
Target	$n{=}1$	$n{=}2$	$n{=}1$	$n{=}2$	$n{=}1$	$n{=}2$
Oscillator	$0.135{\pm}0.005$	$0.127{\pm}0.005$	$0.0922{\pm}0.0034$	$0.0936{\pm}0.0046$	$\mathbf{0.0201{\pm}0.0014}$	$0.0257{\pm}0.0026$
$F_{1}$ coeff.	$0.940{\pm}0.000$	$0.881{\pm}0.000$	$0.512{\pm}0.002$	$0.448{\pm}0.004$	$\mathbf{(3.5{\pm}1.2)\times 10^{-5}}$	$(6.6{\pm}3.7)\times 10^{-5}$
$F_{2}$ high-band	$0.990{\pm}0.001$	$0.906{\pm}0.001$	$0.908{\pm}0.001$	$0.909{\pm}0.004$	$0.921{\pm}0.022$	$\mathbf{(7.3{\pm}3.4)\times 10^{-4}}$
$F_{3}$ burst (%)	$69.5{\pm}0.0$	$69.2{\pm}0.0$	$68.7{\pm}0.0$	$69.0{\pm}0.0$	$69.5{\pm}0.0$	$\mathbf{94.7{\pm}0.2}$

Table 2: Synthetic targets evaluated at depths

n=1

and

n=2

, mean

\pm

standard deviation at the best validation width. Oscillator task reports test RMSE on targets scaled to

[0,1]

over ten seeds,

F_{1}

and

F_{2}

normalised test MSE, and

F_{3}

test accuracy over three seeds (majority-class accuracy

69.5\%

Model	Stores	Discretisation	Block	Spectrum of $A$	Route
S4D ( Gu et al., 2022a )	$a_{k}$ , step $\Delta_{c}$	ZOH	$2\times 2$ rotation	$\Delta_{c}(\gamma_{k}\pm i\omega_{k})$	read-off
S5 ( Smith et al., 2023 )	$\Lambda_{k}$ , step $\Delta t_{k}$	ZOH	$2\times 2$ rotation	$\Delta t_{k}(\gamma_{k}\pm i\omega_{k})$	read-off
S4 ( Gu et al., 2022b )	$\Lambda-PQ^{*}$ , step $\Delta_{c}$	bilinear	dense, diagonalisable	$-2\,\mathrm{artanh}(\Delta_{c}\mu_{k}/2)$	( 9 )
LRU ( Orvieto et al., 2023 )	$e^{-\nu_{k}+i\vartheta_{k}}$	none	$2\times 2$ rotation	$\nu_{k}\pm i\vartheta_{k}$	( 9 )
LinOSS-IM ( Rusch & Rus, 2025 )	$\mathsf{A}_{k}\geq 0$ , step $\Delta t_{k}$	implicit	$2\times 2$ oscillator	$\tfrac{1}{2}\log(1+\Delta t_{k}^{2}\mathsf{A}_{k})\pm i\psi_{k}$	( 9 )
D-LinOSS ( Boyer et al., 2025 )	$\mathsf{A}_{k},\mathsf{G}_{k}$ , step $\Delta t_{k}$	IMEX	$2\times 2$ oscillator	$\tfrac{1}{2}\log(1+\Delta t_{k}\mathsf{G}_{k})\pm i\varphi_{k}$	( 9 )

Table 3: State-space models as depth-one EWS, with the block and spectrum of the EWS generator reproducing each trained model (unit clock for LTI models, accumulated step for Mamba). The

\mu_{k}

are the eigenvalues of

\Lambda-PQ^{*}

, and

\gamma_{k}

\omega_{k}

\psi_{k}

\varphi_{k}

are defined in Appendix E .

Model	EigenWorms	Ethanol	Heartbeat	Motor	SCP1	SCP2	Avg.	Rank
Log-NCDE	$85.6{\pm}5.1$	$34.4{\pm}6.4$	$75.2{\pm}4.6$	$53.7{\pm}5.3$	$83.1{\pm}2.8$	$53.7{\pm}4.1$	$64.3$	$7.5$
S5	$81.1{\pm}3.7$	$24.1{\pm}4.3$	$77.7{\pm}5.5$	$47.7{\pm}5.5$	$\mathbf{89.9{\pm}4.6}$	$50.5{\pm}2.6$	$61.8$	$7.83$
S6	$85.0{\pm}16.1$	$26.4{\pm}6.4$	$76.5{\pm}8.3$	$51.3{\pm}4.7$	$82.8{\pm}2.7$	$49.9{\pm}9.4$	$62.0$	$9.17$
Mamba	$70.9{\pm}15.8$	$27.9{\pm}4.5$	$76.2{\pm}3.8$	$47.7{\pm}4.5$	$80.7{\pm}1.4$	$48.2{\pm}3.9$	$58.6$	$10.33$
DE-SLiCE	$88{\pm}4$	$26{\pm}5$	$74{\pm}5$	$50{\pm}6$	$82{\pm}4$	$50{\pm}3$	$61.6$	$9.67$
D-DE-SLiCE	$86{\pm}6$	$27{\pm}7$	$74{\pm}4$	$55{\pm}4$	$85{\pm}4$	$52{\pm}5$	$63.0$	$7.83$

Table 4: Test accuracy (%) on six UEA datasets, mean

\pm

standard deviation over five seeds, with the average and average rank. Baselines taken from Walker et al. (2024) ; Walker et al. (2025) ; Boyer et al. (2025) ; Terzić et al. (2026) .

PPG-DaLiA
Model	MSE	Params.
NCDE	$13.54{\pm}0.69$	$66$ k
Mamba	$10.65{\pm}2.20$	$797$ k
Log-NCDE	$9.56{\pm}0.59$	$76$ k
LinOSS-IMEX	$7.5{\pm}0.46$	$26$ k
FLASH PD-SSM	$6.5{\pm}0.35$	–

Table 5: Regression and forecasting. Left, PPG-DaLiA test MSE (

\times 10^{-2}

), mean

\pm

standard deviation over five seeds. Right, weather test error at a horizon of 720 steps. Baselines are taken from Walker et al. (2024) ; Rusch & Rus (2025) ; Boyer et al. (2025) ; Terzić et al. (2026) ; Zhou et al. (2021) ; Gu et al. (2022b) . On weather, EWS and D-LinOSS are five seed mean

\pm

standard deviation; others are single seed.

Model	Cycle nav.	Even pairs	Mod. arith.	Parity	Avg.
LSTM	$100.0{\pm}0.0$	$100.0{\pm}0.0$	$99.9{\pm}0.1$	$100.0{\pm}0.0$	$100.0$
xLSTM[1:1]	$53.5{\pm}5.6$	$99.0{\pm}1.9$	$29.3{\pm}1.6$	$100.0{\pm}0.0$	$70.5$
Transformer	$24.4{\pm}0.5$	$90.4{\pm}10.4$	$23.6{\pm}0.7$	$52.2{\pm}0.4$	$47.7$
S4D	$23.7{\pm}1.1$	$68.7{\pm}4.7$	$21.7{\pm}0.4$	$51.2{\pm}1.0$	$41.3$
Mamba	$48.4{\pm}2.2$	$\mathbf{100.0{\pm}0.0}$	$33.1{\pm}6.6$	$54.2{\pm}2.1$	$58.9$
Mamba-3	$69.0{\pm}11.7$	$\mathbf{100.0{\pm}0.0}$	$46.8{\pm}5.5$	$76.5{\pm}7.8$	$73.0$

Table 6: Formal language tasks. Validation accuracy (%) on sequence lengths 40–256, over five seeds for EWS and three for its ablations. Baselines are taken from Walker et al. (2025) ; Terzić et al. (2026) .

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Route	Max. rel. error
S4D	read-off	$9.9{\times}10^{-15}$
S5	read-off	$4.5{\times}10^{-15}$
S4	( 9 )	$2.1{\times}10^{-14}$
LRU	( 9 )	$1.6{\times}10^{-14}$
LinOSS-IM	( 9 )	$4.6{\times}10^{-15}$
LinOSS-IMEX	( 9 )	$5.8{\times}10^{-15}$

Appendix

Table 7: Conversion errors on random parameters in double precision, worst of three seeds. The route is the read-off of the stored parameters, the matching ( 9 ), or the per-channel or per-head matching of a selective model.

Panel of Figure 1 — Figure 1: The EWS block. The clock reads the block’s input, the state of the layer is normalised at each depth separately by $\Gamma$ and read down by $C$ , and both branches are residual.

Dataset	Depth	Level 1	Level 2	Level 3
SelfRegulationSCP1	$2$	$1.0{\times}10^{-4}$	$1.5{\times}10^{-4}$	–
Heartbeat	$3$	$2.9{\times}10^{-5}$	$4.7{\times}10^{-5}$	$6.4{\times}10^{-5}$
MotorImagery	$3$	$2.1{\times}10^{-4}$	$2.9{\times}10^{-4}$	$3.6{\times}10^{-4}$
SelfRegulationSCP2	$3$	$4.9{\times}10^{-5}$	$7.0{\times}10^{-5}$	$8.6{\times}10^{-5}$

Appendix

Table 8: Median relative error of the numerically approximated local EWS against an accurate reference, over ten validation sequences at the initial parameters of each configuration.

Model	Params.	Balanced acc. (%)	AUROC
EEGNet ( Lawhern et al., 2018 )	$0.003$ M	$76.42{\pm}0.36$	$0.8412{\pm}0.0031$
SPaRCNet ( Jing et al., 2023 )	$0.79$ M	$78.96{\pm}0.18$	$0.8676{\pm}0.0012$
ST-Transformer ( Song et al., 2021 )	$3.5$ M	$79.66{\pm}0.23$	$0.8707{\pm}0.0019$
BIOT ( Yang et al., 2023 )	$3.2$ M	$79.25{\pm}0.35$	$0.8691{\pm}0.0033$
EWS, depth one	$0.003$ M	$62.47{\pm}3.48$	$0.6665{\pm}0.0469$
EWS, depth two	$0.020$ M	$79.90{\pm}0.45$	$0.8824{\pm}0.0028$

Appendix

Table 9: TUAB balanced accuracy and AUROC; mean

\pm

standard deviation, over five seeds for the EWS. CBraMod’s balanced accuracy uses a fixed threshold of

0.5

Target $\backslash$ Model	Signature	EFM	EWS
Signature	$(2.00{\pm}0.08)\times 10^{-5}$	$(1.99{\pm}0.08)\times 10^{-5}$	$(2.02{\pm}0.17)\times 10^{-5}$
EFM	$(4.28{\pm}0.53)\times 10^{-2}$	$(2.66{\pm}0.28)\times 10^{-5}$	$(2.77{\pm}0.28)\times 10^{-5}$
EWS	$(2.41{\pm}0.26)\times 10^{-2}$	$(2.43{\pm}0.26)\times 10^{-2}$	$(9.18{\pm}8.80)\times 10^{-4}$

Appendix

Table 10: Expressivity. Test RMSE on targets scaled to

[0,1]

, mean

\pm

standard deviation over ten seeds at depth two.

		$F_{2}$ , normalised MSE		$F_{3}$ , accuracy (%)
$m$	$D$ at $N{=}2$	$N{=}1$	$N{=}2$	$N{=}1$	$N{=}2$
$8$	$72$	$0.939{\pm}0.042$	$0.285{\pm}0.173$	$69.4{\pm}0.1$	$77.2{\pm}10.4$
$16$	$272$	$0.921{\pm}0.023$	$0.046{\pm}0.040$	$69.4{\pm}0.1$	$92.9{\pm}0.5$
$32$	$1\,056$	$0.921{\pm}0.022$	$(7.3{\pm}3.4)\times 10^{-4}$	$69.5{\pm}0.0$	$94.7{\pm}0.2$

Appendix

Table 11: The EWS on

F_{2}

and

F_{3}

by depth and width, mean

\pm

standard deviation over three seeds.

Dataset	Dimension	Observations	Classes
EigenWorms	$6$	$17\,984$	$5$
EthanolConcentration	$3$	$1\,751$	$4$
Heartbeat	$61$	$405$	$2$
MotorImagery	$64$	$3\,000$	$2$
SelfRegulationSCP1	$6$	$896$	$2$
SelfRegulationSCP2	$7$	$1\,152$	$2$

Appendix

Table 12: The six datasets.

Dataset	Path	Gen.	Clock	Readout	$N$	$m$	Batch	Steps	LR	Params
EigenWorms	value, clock	Jordan	clamped	mean	$2$	$8$	$4$	$6\,000$	$4.5{\times}10^{-3}$	$446$
Ethanol	derivative	rotation	time	terminal	$2$	$24$	$8$	$12\,000$	$2.5{\times}10^{-3}$	$2\,508$
Heartbeat	value, base.	rotation	blended	mean	$3$	$10$	$8$	$25\,000$	$3.0{\times}10^{-4}$	$2\,847$
Motor	value	skew	time	terminal	$3$	$16$	$16$	$12\,000$	$4.3{\times}10^{-3}$	$9\,887$
SCP1	value	skew	variation	mean	$2$	$17$	$32$	$25\,000$	$2.5{\times}10^{-3}$	$856$
SCP2	value	skew	blended	terminal	$3$	$14$	$8$	$25\,000$	$5.0{\times}10^{-4}$	$6\,104$

Appendix

Table 13: Selected configurations on the UEA archive. Skew denotes a dense skew-symmetric generator.

	$N=1$		$N=2$		$N=3$
Dataset	Val.	Test	Val.	Test	Val.	Test
EigenWorms	$90.9{\pm}4.9$	$85.6{\pm}4.8$	$\mathbf{96.0{\pm}2.9}$	$\mathbf{93.3{\pm}1.4}$	$93.7{\pm}4.6$	$92.2{\pm}4.1$
Ethanol	$69.1{\pm}13.7$	$66.3{\pm}8.9$	$\mathbf{76.0{\pm}7.7}$	$\mathbf{71.9{\pm}4.8}$	$73.4{\pm}6.5$	$71.4{\pm}3.7$
Heartbeat	$77.1{\pm}3.7$	$76.1{\pm}5.9$	$74.4{\pm}5.6$	$77.4{\pm}2.7$	$\mathbf{77.7{\pm}5.6}$	$\mathbf{80.0{\pm}4.5}$
Motor	$57.2{\pm}3.6$	$50.5{\pm}6.8$	$58.3{\pm}6.0$	$52.6{\pm}7.4$	$\mathbf{66.0{\pm}2.1}$	$\mathbf{61.1{\pm}5.0}$
SCP1	$90.0{\pm}2.5$	$86.1{\pm}3.8$	$\mathbf{90.2{\pm}3.4}$	$\mathbf{87.1{\pm}2.5}$	$84.8{\pm}3.8$	$81.7{\pm}2.1$

Appendix

Table 14: Selected configuration at each depth, mean

\pm

standard deviation of validation and test accuracy (%) over five seeds.

	$N=1\to 2$		$N=1\to 3$
Dataset	Val.	Test	Val.	Test
EigenWorms	$29/0/3,\ +10.7$	$23/0/9,\ +9.1$	$28/1/3,\ +11.1$	$20/0/12,\ +8.2$
Ethanol	$22/1/10,\ +3.8$	$27/0/6,\ +4.2$	$22/0/11,\ +3.4$	$26/0/7,\ +4.6$
Heartbeat	$25/3/5,\ +1.1$	$23/2/8,\ +1.1$	$28/2/3,\ +1.6$	$23/4/6,\ +1.1$
Motor	$25/0/8,\ +1.8$	$20/5/8,\ +1.2$	$28/0/5,\ +4.7$	$19/2/12,\ +1.2$
SCP1	$17/2/14,\ +0.0$	$3/2/28,\ -1.6$	$6/1/26,\ -2.4$	$5/1/27,\ -2.4$

Appendix

Table 15: Paired depth comparison over the

33

configurations of each dataset. Each entry gives the number that improve/tie/fall, followed by the median change in accuracy (%).

Model	EigenWorms	Ethanol	Heartbeat	Motor	SCP1	SCP2	Mean
Log-NCDE	$37\,977$	$31\,452$	$168\,320$	$81\,391$	$91\,557$	$36\,379$	$74\,513$
S5	$22\,007$	$76\,214$	$158\,310$	$17\,496$	$226\,328$	$5\,652$	$84\,335$
S6	$15\,045$	$5\,780$	$6\,674$	$52\,802$	$24\,898$	$26\,018$	$21\,870$
Mamba	$27\,381$	$1\,032\,772$	$1\,034\,242$	$228\,226$	$184\,194$	$356\,290$	$477\,184$
DE-SLiCE	$116\,357$	$16\,964$	$16\,914$	$17\,458$	$25\,154$	$115\,970$	$51\,470$
D-DE-SLiCE	$4\,245$	$1\,796$	$16\,914$	$17\,458$	$2\,402$	$3\,858$	$7\,779$

Appendix

Table 16: Parameter counts at the configuration each paper reports, as published by Rusch & Rus (2025) and Boyer et al. (2025) where available and otherwise computed from the released code. A dash marks a model with no released configuration for that dataset.

Model	EigenWorms	Ethanol	Heartbeat	Motor	SCP1	SCP2
Log-NCDE	$1955/2\,744$	$2116/880$	$785/2\,938$	$779/4\,744$	$616/904$	$586/882$
S5	$11.4/3\,036$	$4.6/1\,430$	$7.5/1\,694$	$23.9/5\,040$	$8.7/2\,462$	$6.2/1\,180$
S6	$25.2/2\,774$	$5.3/1\,012$	$7.3/918$	$38.3/4\,372$	$3.2/1\,178$	$8.7/1\,540$
Mamba	$34.7/3\,786$	$80.5/3\,022$	$37.2/1\,958$	$28.7/3\,498$	$5.2/1\,556$	$31.7/2\,800$
LRU	$22.7/4\,768$	$10.9/1\,406$	$5.4/1\,654$	$38.5/8\,846$	$5.1/1\,140$	$4.7/1\,146$
BD-SLiCE	$10.2/2\,742$	$10.1/1\,136$	$33.5/2\,932$	$28.2/4\,738$	$4.9/892$	$4.8/1\,136$

Appendix

Table 17: Training time in seconds per

1\,000

steps and peak memory in MB.

Model	Params	Time	Memory
EWS	$18\,669$	$8.95$	$1\,722$
D-LinOSS	$301\,196$	$7.97$	$2\,880$

Appendix

Table 18: Weather, parameters, training time in seconds per

1\,000

steps and peak memory in MB, at a batch of

16

Task	$H$	Generator	Clock	Generator	Clock	Smoothing	Params.
		Block 1		Block 2
Cycle nav.	$128$	rotation	clamped	rotation	variation	$0.1$	$92\,219$
Even pairs	$128$	rotation	gated	rotation	variation	$0.05$	$90\,290$
Mod. arith.	$128$	skew	gated	rotation	variation	$0.1$	$96\,165$
Parity	$128$	rotation	gated	skew	variation	$0.05$	$90\,554$

Appendix

Table 19: Configurations on the regular language tasks, using the names of Table 1 . Block one has

m=64

and block two

m=24

, both at depth one.

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Oct 4, 2026, cs.LGJ/K move · Enter open · S save

LogSig-SSM: Time-Series Modelling with Multi-Scale Log-Signature Compression for State-Space Models

Felix Oury, Nicolas Calvo Peiro, Reiko J. Tanaka

Department of Computing Imperial College London · Department of Bioengineering Imperial College London

State Space Models Sequence Modeling

Open paper AskPDF arXiv

Time-series data are often sampled irregularly at high frequencies and exhibit long-range dependencies, which makes long-horizon modelling difficult. Continuous-time models such as neural controlled differential equations (NCDEs) and neural rough differential equations (NRDEs) can handle irregular sampling, but they scale poorly to long sequences. Selective state-space models (SSMs) such as Mamba scale linearly with sequence length, but they provide limited recurrent mixing across hidden dimensions within a single block. We propose LogSig-SSM (Log-Signature Compression for State-Space Models), which first compresses long multivariate time series into a shorter sequence of tokens using multi-scale windowed log-signatures, and then processes these tokens with a selective SSM backbone. LogSig-SSM is scalable and robust to irregular sampling, combining log-signature tokens that capture higher-order cross-channel interactions with a selective SSM that models long-range dependencies. The model also admits a continuous-time interpretation as an NCDE/NRDE-style system driven by a log-signature-based input, in which selectivity induces an input-dependent rescaling of the latent dynamics. Across four benchmarks, namely long-sequence classification on UEA, high-frequency physiological regression on PPG-DaLiA, multivariate weather forecasting, and irregularly sampled clinical prediction on PhysioNet Sepsis, LogSig-SSM outperforms or matches strong SSM and continuous-time baselines while training up to

30\times

faster and using up to

37\times

less GPU memory than Mamba on the longest sequences.

Abstract

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LogSig-SSM: Time-Series Modelling with Multi-Scale Log-Signature Compression for State-Space Models

Reshape and Recur: Improving SSMs with Input Reshaping and Depth Recurrence

A Simple State Space Model Excels at Multivariate Time Series Classification