Random Controlled Differential Equations
Authors: Francesco Piatti
Thomas Cass
William F. Turner
Venue: ICLR 2026 Poster (2026)
Keywords: random features
time-series
path signatures
CDEs
RDEs
reservoir computing
kernels
Primary area: learning on time series and dynamical systems

Abstract: We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping input paths to rich representations. Only a linear readout layer is trained, resulting in fast, scalable models with strong inductive bias. Building on this foundation, we propose two variants: (i) Random Fourier CDEs (RF-CDEs): these lift the input signal using random Fourier features prior to the dynamics, providing a kernel-free approximation of RBF-enhanced sequence models; (ii) Random Rough DEs (R-RDEs): these operate directly on rough-path inputs via a log-ODE discretisation, using log-signatures to capture higher-order temporal interactions while remaining stable and efficient. We prove that in the infinite-width limit, these models induce the RBF-lifted signature kernel and the rough signature kernel, respectively, offering a unified perspective on random-feature reservoirs, continuous-time deep architectures, and path-signature theory.

We evaluate both models across a range of time-series benchmarks, demonstrating competitive or superior performance. These methods provide a practical alternative to explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features. Code is publicly available at: https://github.com/FrancescoPiatti/RandomSigJax.

Published as a conference paper at ICLR 2026
RANDOM CONTROLLED DIFFERENTIAL EQUATIONS
Francesco Piatti, Thomas Cass & William E. Turner Department of Mathematics
Imperial College London
ABSTRACT
We introduce training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping imput paths to rich representations. Only a linear readout layer is trained. resulting in fast, scalable models with strong inductive bias. Building on this foundation, we propose two variants: (i) Random Fourier CDEs (RF-CDEs): these lift the input signal using random Fourier features prior to the dynamics, providing kernel-free approximation of RBF-enhanced sequence models; (ii) Random Rough DEs (R- RDEs): these operate directly rough-path inputs via log-ODE discretisation, using log-signatures to capture higher-order temporal interactions while remain- ing stable and efficient. We prove that in the infinite-widh limit, these models in- duce the RBF-lifted signature kernel and the rough signature kernel, respectively, offering unified perspective architectures, and path-signature theory.
feauure reservoirs, continuous-time deep
We evaluate both models across range of time-series benchmarks, demonstrat- ing competitive or superior performance. These methods provide practical al- ternative explicit signature computations, retaining their inductive bias while benefiting from the efficiency of random features. Code is publicly available at: attps://gite www./raaccecco........... hetett//aannmmmiiseeee
INTRODUCTION
Controlled differential equations (CDEs) generalize ordinary differential equations by allowing dy- namics to be driven by exogenous path [0, 7] + R2 rather than by time alone. This view- point has become central to modern time-series learning: it underlies the continuous-deple limit of residual networks (Cirone et al., 2023), connects naturally to deep state-space models and sequence models with long context (Rangapuram et al., 2018; Gu et al., 2021; Gu & Dao, 2023), and yields the neural CDE paradigm in which the vector field is represented by neural network and learned from data (Kidger et al., 2020; Jhin et al., 2024). Beyond modelling, CDEs provide a clean analytical lens for studying expressivity and invariances of sequence models.
A complementary analytic tool for path-valued data is the signature of a path, i.e., the sequence of iterated integrals that linearizes CDE solution maps in the tensor algebra (Lyons, 1998). Signatures also induce powerful kernels on path space with universality and stability guarantees. In practice, however, signature features and signature kernels can be computationally demanding at high trun- cation levels, motivating approximations and random features.
Reservoir computing offers an appealing alternative: use large, randomly initialized dynamical system to produce rich features of the input, and train only a linear readout (Lukoševičius & Jaeger, 2009). This training-light approach scales well and often works competitively. Yet, principled reservoirs for path data that come with nontrivial statistical kernel limits and that retain the continuous ciiie structure of CDEs are less developed.
1.1 RELATED LITERATURE
A growing body of research has drawn connections between random neural networks and kernel methods. Early works demonstrated that infinitely wide neural networks converge Gaussian pro-

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cesses (Neal, 1996; Williams, 1996). This idea was later extended to deep networks and their train- ing dynamics: in the infinite-width limit, gradient descent on a network is equivalent to kernel regression with the so-called Neural Tangent Kernel (Jacot et al., 2018). In parallel, random feature models were introduced to approximate kernel maps with random projections (Rahimi & Recht, 2007; 2008). For example, random Fourier features approximate shift-invariant kernels with lin- ear readout, while Extreme Learning Machines fix hidden weights and train only the output layer, achieving universal approximation under standard conditions on the activation and sufficient width (Huang, 2014). These insights inspire reservoir computing, which leverages large random dynami- cal systems feature extractors (Maass al., 2002; Jaeger, 2007). This approach scales well and has demonstrated strong performance in time-series forecasting and classification tasks.
For sequence data, the path signature provides an expressive feature map with strong guarantees (Lyons, 1998). spawning a body of work on signature features and signature kernels in learning and statistics (Chevyrev & Kormilitzin, 2016; Cochrane et al., 2021; Salvi al., 2021b; Toth & Oberhauser, 2020). A key advance is the PDE/Volterra characterisation and scalable computation of the untruncated signature kernel (Salvi et al., 2021a), with recent numerical refinements (Cass et al., 2025) and applications across regression, classification, and Bayesian inference (Lemercier et al., 2021a;b).
Randomized signatures compress path information by sampling random linear functionals of the (log-)signature coordinates. Discrete-time constructions with approximation and concentration guarantees developed by Cuchiero et al. (2021), their practical efficacy reservoirs for learn- ing rough dynamics has been demonstrated in Compagnoni et al. (2023), while universal approxi- mation on path space via finite mixtures of randomized signature features was established by Biagini et al. (2024).
Of particular relevance, Toth al. (2025) propose Random Fourier Signature Features (RFSF): the input path is first lifted pointwise into a RBF reproducing kernel Hilbert space via random Fourier features and the signature transform is then approximated in that feature space. We adopt RFSF as a strong baseline and point of comparison for our random differential equation-basee reservoirs.
1.2 CONTRIBUTIONS
In this work, we bridge these viewpoints by developing random feature models for path-valued data that leverage the continuous-tine dynamics of CDEs. Our contributions can be summarized as follows:
Models. We propose two architectures: (i) RF-CDE, which lifts inputs with random Fourier features and then evolves them through a random CDE; and (ii) R-RDE, which operates directly on geometric rough paths via a log-ODE discretization that uses log-signatures to capture higher-order temporal interactions.
Theory. We prove infinite-width limits: RF-CDE converges to the RBF-lifted signature kernel, and R-RDE converges to the rough signature kernel; in both cases, the limiting fea- ture maps induce Gaussian-proees priors over fathffunctionals via the standard kernel-GP correspondence.
Efficiency, At finite width, both models require training only a linear readout, yielding fast, scalable pipelines in the spirit of reservoir computing. We provide user-friendly, optimized JAX implementation, Randomsi gandomsigJax. gJax.
Experiments.
Across time-series benchmarks, our models are competitive with or sur- pass - baselines, including Random Fourier Signature Features.
MATHEMATICAL BACKGROUND
We use the following notation throughout:
V and W denote Banach spaces; (R) is the set of N x N real matrices; C*l (J;V) is the space of continuously differentiable paths from C R into V.
Ar := ts,,t E [0, 0 < st $ T} is the (time) tio-simplex.
EN is the Gaussian measure of matrices in N (R): if the random matrix ~ ENv then its entries Aú are i.i.d. according to a standard normal distribution.
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2.1 CONTROLLED DIFFERENTIAL EQUATIONS
Controlled differential equations (CDEs) describe dynamics driven by a path æ [0, - IR rather than by time alone:
Z,
20
Σ/ fz.)) de
Σ i=1
f: W W.
They capture how system Ze E W reacts to an external control 2, and sit at the heart of continuous- time sequence models.
The main subtlety is how to define the integrals in Eq. 1. If x: has bounded variation - i.e. finite total variation, equivalently finite 1-variation (Definition A.1 in Appendix A) the integrals are classical Riemann-Stieltjes integrals and the CDE is well-posed under standard Lipschitz conditions on fi
If is rougher but has finite p-variation with p 2, Young integration applies and Eq. 1 still makes sense provided the vector fields are sufficiently regular. Beyond this threshold (p 2 2), pathwise Riemann-Stielljes Stieltje///llung integrals break down; we handle this regime by lifting x to a (geometric) rough path carrying iterated integrals and interpret Eq. via rough integration see the rough-path background in Section 2.4.
2.2 SIGNATURE AND SIGNATURE KERNELS
A central tool in the analysis of controlled systems is the path signature, which encodes the essential information of path through its iterated integrals. Let : [o, T + V be a continuous path of finite p-variation with p 2. Then, for any t [), the signature Sig(2) Big(2)o, o, is the unique solution the signature CDE: d dS Sig(z)o, & dres
Sig(z)) 1 (L,0,O....),
taking values in the tensor algebra
T((V)) == {A = (a", ia,,ad al E R, at E v for & 2 1} equipped with componentwise addition and tensor multiplication Explicitly,
Sig(*)o.r = (00))).=((1,)5)()) (z),
5*(x)= Josts
dat, 2 @ dat
SteT
The signature has several important properties, including: (i) it uniquely determines a path up to tree-like equivalence (Hambly & Lyons, 2010); (ii) robustness to missing irregular samples; and (iii) universality, i.c. any continuous functional of a path can be approximated arbitrarily well by linear functionals of its signature.
As the signature is infinite dimensional, for practical use it is truncated. We define the truncated tensor algebra over V of order N E N the quotient T (V) = <((V))/>>N, by the ideal
T>N {A- (a a² ....... T((V)) d =...
0}.
and the truncated signature level N is Sig.V TgN (Sig())) with πgN the canonical projection.
Further details on the tensor algebra are given in Appendix A. and Appendix A.2 reviews the main properties of the signature.
Signature kernels.
Endowing T((V)) with suitable inner product yields the signature kernel
KTtW// (s,
When the inner product on each tensor power 8* k is chosen to be the
inner product
induced from (-, vv. the resulting inner product on subsect of) T((V)) can be defined by linearity as (v. w)r((V)) = Σt=o (Vk, wk) vo). The signature kernel is universal and, when the paths are
differentiable, admits an alternative characterization the solution the linear hyperbolic PDE (Salvi et al., 2021a)
0,0, F.. Sig (s.t) (ic, vtlv K (s,
K K kZ.7(0,t) KI. 0)
(3)
Further details the signature kernels provided in Appendix A.3.
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2.3 RANDOM FOURIER SIGNATURE FEATURES
The RBF kerne admits an explicit feature representation: there is map á IRd - 7 into its reproducing kernel Hilbert space (RKHS) Definition A such that KRBF (2r, y) (o(x), ((v)) Equivalently, by Bochner's 's theorem, any shift-invariant positive-definite kernel k(x, v)) = f( -y) can be written as
/ c/u (2-1) du(w), JRd
k(z, v)
for a probability measure (Gaussian for RBF). This yields Random Fourier Features (RFF): sam- ple frequencies ,.p a and define the real map 60 :R R
of V (e"fur,e).
M (z), (<()) 2 KRBF(I, v).
(4)
where we concatenate the real and imaginary part. When the random Fourier feature map is applied along the path and then the signature is taken we are given the naive Random Fourier Signature Features (RFSF), first proposed in Toth et al. (2025):
RFSFxxF(T) == Sig" (z)) (R2P).
As F grows, & approximates the RKHS feature map of RBF; as N grows, Sig approaches the full signature. Hence taking the inner product (, (x), RFSF approximation to the RBF-lifted signature kernel, which can be defined in the limit as
(B2F) provides a practical
KT. Rig (Sig(ø o 2).: Sig(0 o y)t)T(()))
While computing the full signature suffers from the curse of dimensionality, Toth et al. (2025) devel- ops projection schemes that render RFSFs computationally tractable. Recall that, from a computa- tional standpoint, using explicit features rather than kernlls often avoids constructing and inverting large Gram matrices, yielding far better scalability.
2.4 ROUGH PATHS
Rough path theory generalizes controlled differential equations to paths of limited regularity, includ- ing those with finite p-variation for 2. In contrast to classical paths, rough paths carry additional algebraic structure encoding iterated integrals, which enables well-posed theory of integration and differential equations driven by such paths.
Definition 2.1 (Rough Path). Let 2 1 and let be a control (i.e. w AT + +00) is contin- uous, super-additive, and vanishes the diagonal). A p-rough path over V controlled by is continuous map X Ar - Tiyl (V) such that:
1. x 1 for all s,t 1tc. AT
2. Chen's 's identity holds: Xoo 2 Xyyt Xx,3
3. it has finite p-variation on AT controlled by w, in the sense
Th [X)]lveE _ w(s,t)//
3,((i/D) Vs,t Ar.
Vk I,..., [p],
where Bp IR is a constant that depends only ps and the norm || llver is defined in Eq. 17 in Appendix A.
Definition 2.2 (Geometric rough path). A p-rough path X is called geometric if there exists a se- quence of bounded variation paths (z(")) 1221 such that Tir Sig(z!"))) X in the p-variation metric (see Definition A.3 in Appendix A).
We denote by (V) the space of p-rough paths and by GP, C (V.(V) the geometric ones. Given our definitions 2 (Rd) is the space of d-dimensional continuous paths of bounded variation.
The tensor algebra T((V)) carries the Lie bracket [A, B] := A & B & A. The free Lie algebra on V, denoted L(VV), is the smallest Lie subalgebra of T((V) containing V; elements are finite
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linear combinations of iterated brackets (et, (etg [..., c.,l...., where fei,...,eaf is basis of V. Its degree-m m truncation is (V) == Tgm((((V)
This allows us define the log-signature as log(Sig(>)) L((V) (or log, (Sig(z)) E (V*(V) at finite level), which collects only Lie monomials and thereby removes algebraic redundancies, yielding more compact coordinate system than the raw signature. The logarithmic map acting on T((V)) is described by Eq. 18 in Appendix A.
For 2 and 9 2 1 real numbers, the (rough) signature kernel is the map defined for any two geometric p- and g-rough paths X, Y, by
K x Sig (s, (Sig(X)o,s: Sig( (o,)))(
Finally, rough paths can drive differential equations, which we in this paper. Appendix A.5 provides a brief review, including the Universal Limit Theorem (Lyons, 1998), guaranteeing exis- tence and uniqueness of RDEs driven by geometric rough paths under Lip(7) vector fields (Defini- tion A.5).
RANDOM CONTROLLED DIFFERENTIAL EQUATIONS
In this section, we review the random controlled differential equation model of Cirone et al. (2023), then introduce our variants and state the corresponding limit theorems.
3.1 R-CDE: RANDOM CONTROLLED DIFFERENTIAL EQUATIONS
Let E (); and let DM (0 = to < < tM T"} be any partition of [0, 7]. Consider a I-layer, randomly initialized, homogeneous ResNet driven by æ, with random readout En (independent of the dynamics). Its output is
V (z) = w. Z (x))
where the hidden state evolves by the Euler-type recursion on DM
bu (x) N 1
Ax
N (x)) Art
Z (x) = 20 E RR,,
k=1
with Aut, 2b xt, nonlinearity sp, and i.i.d. random matrices A EN in (R)).
Intuitively, as depth M dc (with mesh size | max;(4+4 - 4)) 0), this recursion
converges to a continuous time controlled system. We take this limit as the definition of the Random Controlled Differential Equation (R-CDE):
az, (x)
Σ A A (z, drl 1=1
(x) = E ERN
(8)
The expected inner product of these features converges the signature kernel, and the readout converges to a Gaussian process with this covariance thereby characterizing the joint infinite- width/continuuus-dege limit of controlled ResNets.
Theorem 3.1 (Cirone et al. (2023)). Let x. y E C² ([0, T): RR: and let Z (x). Z (ÿ) solve Eq. 8 with the same (A:) and ip = id. Then for all 8, [).
N EEN (ZN (z), z (v))
lim N
RN
sig
the (Hilbert-Schmine signature kernel of (æ, v), defined in Eq. 2. Moreover with w ~ En indepen- dent of (A,) and N (x),
lim (z) GP (0, K
N-
in the sense of fintt-dimenns dimensional distributions.
Remark 3.1. Our theoretical analysis assumes Gaussian matrices for simpliciry; however, the clusions hold for any ensemble EN with the standard momenthaill conditions (centered, unit vari-
ance, sub- aab-Gaussian operator-norm tails), shown by Cass & Turner (2024).
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3.2 RF-CDE: RANDOM FOURIER CONTROLLED DIFFERENTIAL EQUATIONS
Motivated by the empirical success of Random Fourier Signature Features (Section 2.3) and by the RBF-lifted signature kernel (Eq. 6), extend the R-CDE framework by incorporating random Fourier lift of the driving signal.
RR be the RFF map in Eq. 4 and, for æt ((,, 17, we denote the lifted Let of Rd path by
x == of (xx) E R.,
where μ is the Gaussian measure. Notice that X is also differentiable it is composition of differentiable functions. Then we define the Random Fourier CDE (RF-CDE) as
dZ...P N (x) dz
Σ A Az
N. (x)) (x)
N... = (o)==
(9)
where 20, (A,) En independent across i and from the RFF randomness, and X, F. denotes the i-th component of the lifted path.
Theorem 3.2. Let Lt. Vt be differentiable paths on [0, T) and Z.N.F(x)) Zi N.F (v) solve Eq. 9 with Ex (independent of the RFF draw). Then, for every 8, t E [0,
= id and the same Ag utd.
lim lim N00
EEN (ZN. .F (.)
K Rig (s. t),
N
where K Sig. RBF (8, t) denotes the RBF-lifted signature kernel (Eq. 6).
We refer the reader to Appendix B.1 for the proof of this theorem.
Gaussian-Process Interpretation. With fixed random reservoir and trained linear readout, RF-CDE implements kernel ridge regression with kernel N (F (a), z,N.F (v)), which verges to the RBF-lifted signature kernel as N, F + co. In this limit, the RF-CDE reservoir defines Gaussian procees-ppcesssseeo prior over funh-funntionals with that signature-basee covariance mirroring the GP limit established for R-CDE in Section 3.1. This provides a clear interpretation of the model's inductive bias: RF-CDE inherits the expressive structure of signature kernels while retaining the scalability of mandom-feature reservoirs.
Discretization. In practice, we discretize Eq. thereby extending its applicability beyond smooth drivers to piecewise-linear paths. We also include bias vector b; Ev together with scaling parameters A, Tb. and oo (uuned via grid search). Applying an Euler scheme yields
A A
N.F (x)
Σ
sp (. (x)) + ophi) AxFF, (0
NNF (x) = 0000 R.V, (10)
VN
where AXF Fri
Fi
F Fi
and x, is the i-th coordinate of the lifted path X
3.3 R-RDE: RANDOM ROUGH DIFFERENTIAL EQUATIONS
We now extend the model to non-smooth drivers by working directly with geometric p-rough paths. This serves two purposes: (i) noisy time series often benefit from higher-ordes information, which signanures/oog-sinnauurs provide stable features; (ii) in many applications the (log-) tiignarre is already available or estimable, so operating in rough-path space avoids information loss.
Let HVm(V, W) be continuous linear map. For each i 1. induces map sk vok wo
pook (v1
3 vx) f(v1) x f(()) with
y0u Id.
The elements of &ok C T((V) can be interpreted as functions on words of length k over an
alphabet A = 1,... d. A word w indg...i4 corresponds to the basis element e₁ ... @ es2
in VSk Denote Wm the set of all words formed by letters in Ad of length |w] < m, and

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Wa = Um2o WT Let (A,) End(RR) (the algebra of endomorphisms of R under composition with unit Idy), and let A'A T(R))) - EId((R-)) be the unital algebra homomorphism
1 A,, ... Ain for w = i1 ii.i i E Wa, (11)
Ga( = Σ (G, w) where Aw =
N
wEWa
extended multiplicatively by [a(wv) = - o (v)) and [A(1) = Idy. Here (G, w) denotes the wo-ccordinate of the tensor G ). The map LA is well defined on group-like tensors G(R))) C ((RR)) (see Appendix A.4) and in particular on signature increments of geometric rough paths; we refer Lemma 6 in Appendix B.2 for the precise statement and proof.
Linear development along a path. For a -varittinn path x : [0, [] + R set
(1( == I'A( (Sig(z)o.)) E End((R)),
(12)
Lemma Let a' be as Eq. 11. Ifx= : [0, T] - Rdd has bounded variation and at. E Th with A; E End(RR)) then StA in Eq. 12 is the unique solution of the linear CDE
=
(x) = S o dat
()) = I E End((R
(13)
See Appendix B.2 for the proof.
By continuity of the Itô- ItonLy map (Theorem A.5), Eq. 13 extends to geometric rough drivers. If X E (R*), there exists a unique matrix-valuee geometric p-rough path S GD2 (End(R") given by the canonical lift the solution to the rough linear equation
ds (X) d%,
(z) Idy E End(R
(14)
ie the first level of the lift: Tı (S..1) == 5 E (R)
Random RDE features. Let (A,) ~ be the valued geometric p-rough path associated with Eq. 14. Define the one one-form
iid.
EN be Gaussian random matrices in Mx (TR), and let S4
R:R. N (H) ),
{())[M]
M((())),
with non-linearity &o E Lip(7) (Definition A.5) and > p. The andom-feature path 2 ZN(X) [0,2] + R is then defined as the unique solution of the Random RDE
dz,((x) (x) S(ZN) ds
2 50 ERN,
(15)
where 20 EN is independent of {A,]
Remark 3.2. In the smooth case (so X = Sig(z) with of bounded variation), Eq. 15 yields dz,- Di= A, (Zz)) dæt. A brief derivation is provided in Appendix B.2.
We now state and prove two theorems: the first establishes existence and uniqueness, while the second shows convergence to the rough signature kernel introduced in Section 2.2.
Theorem 3.3 (Existence and uniqueness). Let X E (RR) with p 2 1 and yp E Lip(r) with > p. Then the R-RDE 15 admits a unique solution z E ([ ), and the Itô ItonLyons map (X, 20) + Z continuous the rough-pan topology.
Proof. This is a direct application of the Universal Limit Theorem (Theorem A.5) under Lip(n) vector fields (Definition A.5) with 7> p.
Theorem 3.4. Let X E (R²) and Y (Rd) be geometric rough paths. Let Z (x) and Z (Y) be the solutions of Eq. 15 with y id and the same matrices (Ar)e (with A, EN i.d.d). Then for all s, t [0,
lim
BEN (Z), 2, (Y)) = K (x, (s, f),
No00
where where KX denotes the rough signature kernel defined in Eq.
We refer the reader to Appendix B.3 for the proof of this theorem.
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Gaussian-process interpretation. Analogously to the RCDE and RF-CDE settings, fixed R- RDE reservoir with a trained linear readout induces a Gaussian- process prior over path functionlsss, with covariance given by the rough signature kernel K Sig
Log-ODE discretization. For rough drivers, naive Euler schemes ignore the algebraic structure of the signal, breaking Chen's 's multiplicativityy The log-ODE method (Lyons, 2014) addresses this by summarizing each time step [bi, (2+1 via the log-signature
L = logm (Xissten)) (R")).
which maps increments into the step-m free Lie algebra see Section 2.4. One then advances the state on [t,, t4+1] by solving an ODE with constant Lie coefficients L, This preserves the group/Chen structure exactly and removes the algebraic redundancies of the tensor algebra.
Let W be fixed Hall/Lyndon basis of Lie words of length |m|| matrices (B....., Ba} with B, iid.
Given collection of random EN. define the linear map 1I _ (Rd) + End(R by
(C) = Σ
B(w) (logm (Grasta)ss))
B(w)
No .... [ Bigb, Bigl ..... Bal
WE War
for w = i1 ... iss where [A, B] = AB BBA is the Lie bracket.
Given X E GD2 (IE) and partition Dis {0 to < < tM T}. we update
2, N, 1I
II (X,,
(
N 20 R.N.
(16)
which uses only the log-signature coefficients (logm. (Ki,, ), and the corresponding com- mutators B(w), keeping the discretization faithful the rough-path algebra while remaining ex- plicit. Scaling and bias hyperparameters are incorporated analogously to the RF-CDE discretization (Eq. 10) and tuned by grid search.
While more computationally expensive than RF-CDE, R-RDE can be advantageous very long sequences: the log-ODE discretisation operates on log-signatures computed over chunks, thereby shortening the effective trajectory length fed to the random differential equation.
4 EXPERIMENTS
In this section, we detail our random differential equation models' performance for time series clas- sification. We implement RF-CDE via the discretization in Eq. 10 and R-RDE via the log-ODE scheme in Eq. 16. For completeness, we also benchmark the R-CDE of Cirone et al. (2023) described in Section 2.1 which, to our knowledge, has not been tested.
Benchmarks. We compare against Random Fourier Signature Features (RFSF) in the two projec- tion variants of Toth et al. (2025) - Diagonal Projection (DP) and Tensorized Random Projection (TRP). We also benchmark the PDEEbased signature kernel (SigPDE) with the RBF base kernll (Salvi et al., 2021a). and standard time-series baselines: Random Fourier Features (RFF), RBF, GAK, and Random Warping Series (RWS). For space, full results for RBF, GAK, RWS, and RFF are deferred to Appendix C, as these methods rarely attain state-of-the-an performance on our suite but are included for completeness. We do not include the truncated signature kernel (Király & Ober- hauser, 2019) in our benchmarks, as its exponential feature explosion makes it impractical in all but small and low- di-ddmmensionll datasets. Neural benchmark such as Neural CDE (Kidger et al., 2020) and Neural RDE (Morrill et al., 2021) have also been included in the ablation study of UEA time series classification and in the Hurst-exponent classification experiment.
Remark 4.1. SigPDE, GAK, and RBF are not random-featur methods: in the SVM setting they require computing and inverting the kernel Gram matrix, which can be bottleneck the number of samples grows. By contrast, mandom-featuee models learn only a linear readout on top of fixed random dynamics, avoiding kernel matrix operations and retaining linear-in-sample complexity.
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Computational Time. Table 4 summarizes the computational complexity of our signature- random-featune extractors. These models scale linearly with the sequence length e, unlike kernel baselines such as SigPDE, RBF, and GAK which scale quadratically. Among methods, R-RDE typically the slowest due to an additional O(N3) component arising from matrix development; however, this cubic term is independent of the batch size B, it can be precomputed.
RESF-DP O(BeN'd) O(BEF(N" + d)) O(N (BL+ ( O(BEF(Md 2² O(BEM (dF F²))
R-CDE
RFCDE (ours)
R-RDE (ours)
RESF-TRP
Table 1:
cost (ignoring the final linear readout). Here B is batch size, F the number of random Fourier features, d the input dimension, l the sequence length, M the truncation level the signature, and N the (output) feature dimension.
Asymptotic
4.1 UEA TIME SERIES CLASSIFICATION
UEA Datasets. The UEA archive (Dau et al., 2019) is collection of datasets for benchmarking classifiers on multivariate time series classification problems. The data modality ranges from various sources e-g. human activity recognition, motion and ECG classification, audio spectra recognition, and others. A summary of the dataset characteristics can found Table 2 in Dau et al. (2019).
Experimental Sctup. Full details, including grid-search ranges, are deferred to Appendix C.I.
Results. With 250 random features, RF-CDE the strongest of caddom-feature models on av- erage, followed by the non-parametric SigPDE baseline. Averaging across the 16 UEA datasets, RF-CDE attains 0.741 accuracy versus 0.738 for SigPDE, while RESF variants are slightly behind (0.725-0.726), and R-RDE and R-CDE trail further (0.708 and 0.695, respectively). RF-CDE is particularly competitive on medium difficulty tasks (c.g., Libras, NATOPS), and R-RDE occasion- ally leads among random-feature methods structure-rich datasets (e.g., U'WaveGesturelllbrayy These trends suggest that injecting an RBF lift before the controlled dynamics (RF-CDE) is an ef- fective way capture local geometry in continuous time, whereas the rough-path variant (R-RDE) helps when higher-order interactions matter. Table 4.1 reports the results.
medium-difficulty
R-CDE
RF-CDE
R-RDE
RESF-DP
RFSF-TRP
SigPDE
Articulary WordRecognition
0.950
0.967
0.957
0.977
0.983
0.983
AtrialFibrillatio
0.333
0.467
0.467
0.400
0.266
0.333
BasicMotions
1.000
1.000
1.000
0.975
0.975
1.000
Cricket
0.972
0.972
0.902
0.972
0.958
0.972
EigenWorms
0.420
0.630
0.612
0.786
0.771
0.794
Epilepsy
0.935
0.971
0.935
0.942
0.942
0.891
RthanolComcentrratoon
0.312
0,358
0.373
0.430
0.407
0,460
FingerMovements
0.550
0.550
0.530
0.570
0.530
0.610
Handwriting
0.331
0.331
0.331
0,380
0.362
0.409
Libras
0.889
0.911
0.867
0.833
0,906
0.867
NATOPS
0.872
0.944
0.906
0.889
0.906
0.928
RacketSports
0.809
0.809
0.737
0.829
0.809
0.849
SelfRegulationSCPI
0.877
0.877
0.840
0.887
0.904
0.904
SclfRegulationSCCF2
0.555
0.555
0.567
0.483
0.494
0.544
StandWalkJump
0.467
0.667
0.400
0.400
0.533
0.400
UWWeeGesttreLLibrayy
0.853
0.844
0.903
0.856
0.853
0.866
Avg. acc. (t) Avg. rank (L)
0.695
0.741
0.708
0.726
0.725
0.738
4.250
3.062
4.125
3.406
3.594
2.562
Table 2: UEA test accuracies with N = 250 signature-based aaddom-feature models (SigPDE is kernel baseline: no random features). For each row, the best result highlighted in bold.

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Ablation: Number of Features. We repeat the full protocol with 64 and 500 random features. In the low-budget setting, RF-CDE performs particularly well relative to the RFSF and neural base- lines. On the other hand, doubling the feature budget yields modest gains typically few per- centage points on the more challenging datasets - while leaving easier tasks essentially unchanged. Being kernel methods, SigPDE is unaffected by this ablation. Results are included in Appendix C.
4.2 CLASSIFICATION OF ROUGH SIGNALS VIA HURST EXPONENT RECOVERY
To further assess the ability of our model to extract fine-grained geometric information from irregular time-series data, we introduce a controlled classification task based on synthetic fractional Brownian motion (fBm). Each sample consists of fBm with Hurst exponent H E {0.05, 0.15, ..... 0.755, and the goal is to correctly identify the underlying Hurst parameter from the observed path. Recall that H controls the roughness of the process through its p-rriiaiion.
We evaluate two variants of the task. V1 uses the raw fBm trajectories. V2 applies per-sample standardisation (zero mean and unit variance), thereby forcing the models to exploit only geometric features and long-range dependence. Full details of this experiment are deferred to Appendix C.S. Table 3 reports classification accuracies across range of competing baselines. In both settings, R-RDE model achieves consistently strong performance, and in the most challenging regime (V2, N 64), preserves clear performance margin over alternative approaches.
Table 3: Hurst-exponent classification accuracy. Here denotes the feature dimension of rantorefeature models (with neural baselines matched in parameter count).
Setting
R-CDE
RF-CDE
R-RDE
RESF-DP
RESF-TRP
NCDE
NRDE
V1 N 64
0,870
0.895
0.955
0.840
0.895
0.905
0.920
V1 N 100
0.900
0.945
0.950
0.890
0.910
0.895
0.945
V2 N 64
0.635
0.645
0.735
0.630
0.650
0.650
0.675
V2 N =100
0.650
0.695
0.730
0.675
0.675
0.650
0.685
4.3 ROBUSTNESS TO MISSING DATA
To assess the robustness of our models to incomplete observations, we perform an additional ex- periment multivariate time series from the UEA archive. We synthetically introduce missing data by randomly removing individual time points along the test trajectories. The experiment de- scribed in Appendix C.4, and the corresponding accuracy tables are provided therein (Tables 7 and 8). The results show that our models remains competitive even under substantial information loss: in particular, RF-CDE exhibits the most stable performance as the missingness level increases.
CONCLUSIONS
We introduced training efffccient framework for time-series learning based random continuous- time reservoirs whose infinite-width limits coincide with established path kernels: RF-CDE yields the RBF-lifted signature kemee,, and R-RDE yields the rough signature kernel. This places models on the same framework as infinite-width neural networks. Empirically, with only a few hundred features, both models are competitive on UEA benchmarks while avoiding kernel-matri inversion and scaling linearly in sequence length. The result is a scalable alternative to explicit signature computation.
Future Directions. Looking forward, we natural extensions: learn (or sparsify) the spectral measures that define the reservoirs, and couple our continuous-time features with probabilistic heads for calibrated uncertainty and streaming inference. It will also be interesting to study NTK dynamics around the random reservoir, to design adaptive log- lorustrue discretizations for very long contexts.
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ACKNOWLEDGMENTS
TC has been supported in part by UK Research and Innovation (UKRI) through the Engineering and Physical Sciences Research Council (EPSRC) Programme Grant: UKRI1010 High order mathe- matical and computational infrastructuree for streamed data that enhance contemporary generative and large language models. FP and WFT have been supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1), ChatGPT-5 was used by FP to refine clarity and style of selected paragraphs. For the purpose of open access, the authors have applied Creative Commons Attribution (CC BY) licence any Author Accepted Manuscript version arising.
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Thomas Cass and William F Turner. Free probability, path developments and signature kernels as universal scaling limits. arXiv preprint arXiv:2402.12311, 2024.
Thomas Cass, Francesco Piatti, and Jeffrey Pei. Numerical schemes for signature kernels. arXiv preprint arXiv:2502.0044 2025.
Ilya Chevyrev and Andrey Kormilitzin, A primer on the signature method in machine learning. arXiv preprint arXiv: 1603:0:733300 2016.
Nicola Muca Cirone, Maud Lemercier, and Cristopher Salvi. Neural signature kemells as infinite- width-depth-limits of controlled resnets. In International Conference on Machine Learning, pp- 25358-25425. PMLR, 2023.
Thomas Cochrane, Peter Foster, Varun Chhabra, Maud Lemercier, Terry Lyons, and Cristopher Salvi. Sk-tree: systematic malware detection algorithm streaming trees via the signature kernel. In 2021 IEEE international conference on cyber security and resilience (CSR), pp. 35-40.
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Albert Gu, Karan Goel, and Christopher Ré. Efficiently modeling long sequences with structured state spaces. arXiv preprint arXiv:2111.00330 2021.
Ben Hambly and Terry Lyons. Uniqueness for the signature a path of bounded variation and the reduced path group. Annals of Mathematics, pp. 109-167, 2010.

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A ALGEBRAIC AND ANALYTIC BACKGROUND
This appendix collects the algebraic objects (tensor and free Lie algebras, group-like elements), the analytic objects (signatures, rough paths), and the key theorems rely (the PDE char- acterization of the signature kemel and Lyons' Universal Limit Theorem). We keep statements self-contained; detailed proof' can be found standard references (Cass & Salvi, 2024; Lyons, 1998).
A.1 TENSOR ALGEBRA
Let V be a Banach space. The spaces of formal polynomials and formal power series over V are defined respectively as
00 V
80
T(V)=
and
T((V))
II vok
where V denotes the k-fold tensor product V. Both T(V) and T(()) can be endowed with the operations of component-wise addition and multiplication @,, the latter defined for any two elements (oo, 01, o1,...) and IB = (bo, bi....) as
& a & ba-is Vk 2 0.
A ② B (co: C1, C2,
where
V 3 ct
2=0
When endowed with these two operations and the natural action of R by AA (Aag, Aa T(())) becomes real, non-commutative unital algebra with unit 1 = (1, 0,0, O,...) called the tensor algebra.
The level-m truncated tensor algebra over V of order N is the quotient
m
(T = -(V-)(((V)))))))))) (V)
v@k
k=
where
TVm(V) = {A = (a a ........) E T(()))
=0]
We denote by Tmm T((V)) + T (V) the canonical projection and by क T((V)) V the level maps.
Finally, we can define a norm on V as
(us oiv,
for 02 Up-
(17)
A.2 SIGNATURES AND THEIR BASIC PROPERTIES
Signatures extend iterated integrals beyond smooth curves, but to do so we must quantify how "rough" path is. The right scale p-variation; it measures the cumulative oscillation of a path and yields topology under which Young integrals (for < 2) and, more generally, rough integrals (for p 2 2) are well posed.
Definition A.1 (p-variation). Let [0, + V be continuous. For any [s, f] [0
1/9
| |Ir||p-varr[s.]]
sup DC[..1]
Σ
||atern - 2z,
LED
where the supremum is over all partitions D of [s. (].
The induced p-variation metric on C((0, C[OO.T]:V] T):V) is
var (I. y) 10 sel--vlllaaee yllp-var.(o.TT)
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Let x: [0, + V be path of bounded variation. Its signature over [s, is
Sig(æ),, (1, (1.5)
(*) J
dr @ duu..
The signature satisfies Chen's identity (it is multiplicative):
Sig(z)..u & Sig(z),.: = Sig(
Lemma (Factorial decay), For x of bounded variation and all & 2 1.
dru, @ ... 8 dous
k!
Vor
where the norm on Vosk is given by Eq. 17 and [[avar(aed) denotes the 1-variation norm given by Definition A.I.
Three foundational properties make signatures effective for learning:
Theorem A.1 (Uniqueness up tree-like equivalence). If Sig(t)), Sig(y)o, then and are tree-like equivalent; conversely, tree-like equivalent paths have equal signatures. On classes where tree-like structure is ruled out (e.g. reduced paths), the signature injective.
Theorem A.2 (Universality of linear functionals on signatures, (Hambly & Lyons, 2010)). Let K. be a compact set of bounded-variation paths (modulo tree-like equivalence). Then the linear span of coordinate iterated integrals ((,,Sig()) Sig(.)o.r) continuous functions on K. with the topology of uniform convergence.
T(())) finite) is dense C(C),, the space of
invariance). Let x. [to. T] - TR. be continuous path of bounded variation and let (a, b] and [c, be two subintervals of [to: T1. Let X [c, d] + [a, b] be a reparam- eterization. Then Sig (x) az. S(z A)e,d-
Lemma 3 (Reparameterizatio
For some applications it might be important to keep the time parameterization of the path x. In this case, suffices add time as an extra coordinate to get the time-augmented path : I +> (1, 2x).
A.3 SIGNATURE KERNELS AND THE PDE CHARACTERIZATION
Kernel methods measure similarity via inner products a (possibly space. This is formalized by the notion of a reproducing kernel Hilbert space (RKHS): Hilbert space of functions in which point evaluation kernel. We recall the standard definition below.
feature
continuous linear functional represented by the
Definition A.2. Let X be nonempty set and k be positive definite kernel on X. A Hilbert space Hy of real-valued functions on X' equipped with an inner product (-,-Hy, is called a reproducing kernel Hilbert space (RKHS) with reproducing kernel k, if for any I X and for any f E H2 the following two conditions are satisfied:
1. the feature map k(x,,) -) E Hai
2. the reproducing property f(x) (f,k(z, H), holds.
Intuitively, the element k(x,,) He plays the role of an (often (iinii-ddimensoonaa) feature vector for An immediate consequence of the reproducing property is the feature space inner product identity
k(r, v) (k(x,) :), k(y, H)
HeN
y E X
In setting, the feature map of path is its signature the sequence of iterated integrals living in inner product (1 vy on V, define for 2
the tensor algebra. Given an
(v) ... Uk. W1
2 wk) vox
II (ujs wylv.
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For & (ao, .1....) and B (bo. bi....) in T((V)) set
(A, (A,
(akr be.bb))an
k=
This induces the signature kernel
Sig (s, = (Sig(z)..s, Sig(y)o+)r(v>>>
When are differentiable, K is characterized as the unique solution to linear hyperbolic Goursat PDE.
Theorem A.3 (PDe/Volterra characterization, (Salvi et al., 2021a)). Let (²O(0,T);)) Then k(s,t) = Kiig (s, t) solves
0, = (i0.36)v k(s,t) k(s, = k(0,t) 1,
equivalently,
k(s,t) = =1 1+ 1.. (ihu bolv k(u, v) de du.
Conversely, the (unique) solution of this problem coincides with (Sig(z)o,.: Sig(y)o.))-
direct corollary is a universality statement for the induced kernel on path space.
Theorem A.4 Ohniverslityyccanntttiiiiiees of the signature kernel). On compact of paths (modulo tree-like equivalence), ), the signature kernel is universal (its RKHS is dense in continuous functions) and characteristic (mean embeddings of Borel probability measures are injective).
A.4 ROUGH PATHS AND LIE ALGEBRAS
For tensor-valuce paths, the p-variation extends in the usual rough-path way: apply p-variation level- wise each homogeneous tensor component and combine (up an equivalent norm) obtain the standard p-variation metric T(pl (V).
Definition A.3 (p-variation metric). The p-variation metric of two p-rough paths X, Y Ar + Thel (V) is defined as follows
1/9
d Y)
max sup Iskšipl
Σ
p/k (Xe,, The (7, 1|
MED
where the supremum taken over all partitions of the interval [o, T7 and the norm on Vo is given by Eq. 17.
This metric is the natural topology for rough paths (Definition 2.1): the class of geometric p-rough paths (V) the closure, under dy. of truncated signatures of bounded-variation paths (Defini- tion 2.2).
The associative tensor algebra (T((V)), 60) carries commutator
[A. & F B & A.
Equipped with [. the same underlying vector space becomes a (noncommutative) Lie algebra. Iterated brackets quantify non-commutativity and generate the "Lie words" that will organize the algebraic content of iterated integrals.
Definition A.4 (Lie polynomials and Lie series), Let Lo 0, V, and L [V, with UV denoting the linear span of all elements of the form [e, fl where (e, V x for any two linear subspaces V. U of T(V)).
The space of Lie polynomials over V, denoted as L(V), is defined as:
L(V) =
L
R=
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The space of Lie formal series over V. denoted [((V)) T((V)) is defined as Vk 2 0,1 L
(((V)) = (-=
For any 2 1, the step-n free Lie algebra is defined by (V) = Ten (C((V)) with elements called Lie polynomials of degree Define the (formal) exponential and logarithm w.r.t tensor multiplication,
20
A &n
m-
e
log( 1 + A)
=
on
(18)
n!
n
n=1
and their level-m truncations expm = 5mm o exp, logm == T5mm log.
The group-like subset
G(V) = exp (E((V))) c T((V))
Lie group under the tensor product; level write G" (V) Tg (G(V)) with mutually inverse maps
exp: C((V)) - G(V)
and
log G(V) + L((V)),
and
exp, (V) - (V) and For any bounded-variation path (and, by continuity, for geometric rough paths), the signature Sig(z) G(V) is group-like. Correspondingly, the log-signature log(Sig(z)..)) lies in the free Lie algebra [((V) (or its truncation (V) at finite step). This identification what allows Lie- algebraic discretizations (e.g. log-ODE schemes) that respect the path's 's multiplicative structure.
loga (PVV++="00
A.5 ROUGHI DIFFERENTIAL EQUATIONS (RDEs)
This section follows Cass & Salvi (2024) and gives precise pathwise meaning to rough differential equations (RDEs)
dY, = (V)) Yn = E N
(19)
driven by a geometric rough path X (Definition 2.2).
The main idea is approximate the (geometric) rough path X (V) by signatures of smooth/bounded--vrrittooo paths in p-variation, solve the ordinary controlled differential equations (CDEs) along those smooth drivers, and define the rough solution as the uniform limit of the smooth solutions. The key tool is Lyons' Universal Limit Theorem (Theorem A.5), which also yields sta- bility/continuity of the solution map (the Itô -Lyons map).
Before stating it we recall the definition of --Lipsccitz function (in the sense of Stein).
Definition A.5 (Lip(>) functions). Let V, W be two normed space and let y > 0. A function 9: V + W is called --iiscchitz if g is Lr] times continuously differentiable and such that there exists constant M 2 0 such that the supremum norm of its ktht derivative, k = 0,.... lty). and the (-- - (>)l-]) -rod)-Hölder norm of Ln)th derivative are bounded by M. The smallest M satisfying these conditions is the Lipschitz norm of g. denoted by ||gl| := 11glleip [1olleip(vv.W) (V.W). We denote by Lip (V.W) the space of --Lisschitz, functions from V to W.
One-forms and the rough integral (informal). In an RDE the map f : W - Hom(V, W) is naturally viewed as a one-form: each state y W. ( is linear map V W be integrated against the (rough) increment of the driver. When V = Rd we often write f = (Ji...,, fa) with fi : W. so that the coordinate form is
dY, Σ (X)) 1=1
Since is rough, the integral cannot be defined by Riemann Stinann-Stieltjes sums at level I only. Instead uses all available iterated integrals of X up level [p]-
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Controlled paths and compensated Riemann sums. A path Y [0, W is controlled by if its increments admit an expansion Yu,, = Y, % + Ru, with higher-order consistency and suitable p-variation bounds on the remainder Ru,.. For such and f E Lip(7) with 7 > p, the rough integral So s(Y.) dz is defined as the limit, as (D) 0, of the compensated sums
Σ (( + D((Y))O(.)) X ... D (OO)) M.DEED
((Y.) x
m-- times
where X u,p E V@k are the level-k increments of the geometric rough path. This construction agrees with the classical Riemann Stieltjes (or Young) integral when the driver is smooth (or has 2), and it is the pathwise notion used in Eq. 19.
Shemann-Stieltjess
Theorem A.5 (Universal Limit Theorem yonos, 1998)), Let 2 and let X [0, T - be sequence of continuous paths of bounded variation which converges in p-variation to geometric p-rough path AT + TiP((V) Let f V + Hom(V, W) be a Lip(7) function with 7 > p. Consider the controlled differential equations
dY, dXX")
Yo = E W
(20)
Then, there exists unique geometric rough path Z (X, Y) Ar Tlpl @W) such that Y converges to in p-variation. Moreover, the Itô map I (yo. X) + Y is continuous in p-variation.
Definition A.6 (RDE solution). Let X GV2 be a geometric p-rough path. We say that the continuous path Y [0, T] + W of finite p-variation is solution to the RDE
dY, f (Xi dxa, Yo = yo W
if Y belongs to the set of (uniform) limit points constructed in Theorem A.S. In particular, if f W HV W) is linear or L-pischhitz with y p. then Y is unique.
The notion of RDE solution presented in Definition A.6 maps a geometric p-rough path to a W- valued continuous path of finite p-variation. However, might desirable to construct "full" solution also as geometric rough path. This is the case, for example, if one is interested in using solution to a first RDE to be the driving signal for a second RDE. More precisely, we will say that Y E GG((W is the (full) solution to the RDE
d', = f (Y,) dxe, started at Yo E Sig (S2(WW)
there exists sequence (X") of continuous bounded variation paths such that the sequence of truncated signatures (Sig [P] (X*)) converges in p-variation to and such that the sequence (Yo- Sig Il (Y")) converges uniformly on [0, T) to as n + co, where {Y"} are the solutions to the CDEs 20, with Yo (%).
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B PROOFS
Before proving the main theorems, we record two auxiliary results used in our proofs. First, we show that the inner product of time-derivatives of Random Fourier Feature lifts converges almost surely to the corresponding RKHS cross term; we include a self-contained proof. Second, we invoke trace-moment identity for products of random matrices from Cass & Turner (2024) (proof therein).
Lemma 4. Let Lerx, x, E (10, T], R24) and o" 12d + IR2F be the random Fourier feature map
(2) V cos(w]) 2), sin(w] 2), .... cos(w)22), sin(w]2)) R2
where {wo) lid μ and is the standard Gaussian measure on R. Define the lifted curves
x = 6% (xt);
and YP == ==
Let : (1: RR) and define the (infinite-din (imensioadmm feature map
(u, v) / u(w) T v(w) du(w).
6(2) = (cos(w" T2), 2), sin(wT IR?,
Jzd
Let Xx = o(x,) E H. and Y; = (lr) E Hi then, for every 8, 1 E [0, Ty. we have
1. almost sure convergence
(X,, Y.F
as
(x.: Vi)x
(21)
/2FF
F-000
2. convergence in L.=:
So J' 188, Y:")
(Xa, Yi)xl dl ds F-00, 0.
(22)
where X is the derivative w.r.t time of X and similarty for Y.F, Xe. and Yv.
Proof. Differentiating component-wis wise.
dt cos(w] 21) = sin(w] rz) W] in.
dt sin(w] xx) = cos(w] xx) [e] do,
and similarly with Fr, dt replaced by Vts jt. Hence
XF F 1 ( sin[(w]] x.) wJ te, cos(w] w)) its p
VF
j=1
and analogously for Y.F The Euclidean inner product becomes
(XF
Σ [w] it (w) in.) (iin(w] x.)) sin(w] we) + cos(w] xx) cos(w]] vi)
F
Σ
[w] tt)) (w)) (n) cos (w]] (x. -v)) =
1
Σ
Dws (8,
F
j=1
f=1
Define
(s, t) (w) ūn) cos (w T (xr 41)).
Then (s, f)1f are i.i.d. with mean
(s, / (wT (wT (w) it) cos TwT
(s., - y.)) dy(ww).
On the other hand, in the RKHS model H 1.2 (,2(
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Xe(w) = (- sin(w x2) wT 2.. cos(w T xx)) wTTTa iz),
and similarly for Ye(w). Therefore
(x..t)) / (uTa))(s ue) sin((w] T 2.)) sin(w ve) + cos(w] xx) cos(w ue) du(w)
( (w tw un) cos (w (wx ue)) dp(ww (x
Thus Ex( [)] (x., vi)x- By Bauchy-Schwarze and cos S
l t) & ||]l | l
Since
the sequence {mw, (s,() is i.i.d.,
ii v, y E C² ((0, 7), (R") implies that |dt.l1, /e are bounded on [o, T).
ii.. f [|]| du(w) < oo as the Gaussian distribution has finite variance, so g (8, is inte- grable.
we can apply the strong law of large numbers, giving that
F Bus (s,
Σ
as.
Ex(gw(s,t)) = =(x (8 x1)
F
F-000
==1
which proves Eq. 21. For the L convergence,
1 F
Σ [k,])l²)
||
lbilll
uss. F-
Ep
| 11
B2F
Since En |w]l < oo for Gaussian μ4, there exists C(w) < oo such that, for all F,
1Xx )gar| & C(w) || lleell
and
((xx, Yo) s C(w) || ledel
Hence,
{xx, v.F) ) p2r - (X,, v1) (0(w)
|| ||bel
(23)
Since à. L' L'O, the envelope on the right is integrable over [0, 7J2. Combining the almost- sure pointwise convergence with the bound in Eq. 23, and using the continuity (hence measurability) of the maps
(s.t)
and
(s,
Vi)
BaF
the hypotheses of the dominated convergence theorem satisfied. Therefore,
[/ 1X5, RaP - (Xx, Yi)ml ds F-200 O,
which proves Eq. 22.
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Lemma 5 (Cass & Turnrr (2024)). Suppose that for each N N, iiis d is col- lection of d Gaussian matrices. Let n, m 2 0 be non-negative integers and consider the words I = 1. in Wa and J Im E Wa (where Wa is defined in Section 3.3), with corre- sponding matrix products
A = AÑ A.
AW. = (AP)
T Ay
Then, setting h: n + m,
lim N
1 E tr (AWr)) 2
{0,
1, iI=J otherwise
We use the convention that if n = 0. then AÑ = Idy, the x N identity matrix.
B. PROOF OF THEOREM 3.2
For convenience we restate Theorem 3.2 and provide its proof.
Theorem. Let Tt, Vt be difserentiable paths on [0, T] and 7.NF (x), Z.N.F (y) solve Eq. 9 with y utd. id and the same As EN (independent of the RFF draw). Then. for every s, E [0,
lim F N
lim
BEN (2 (2) (z), ZIN. (v)) (NN)(()))
KEy Rig (s, t),
where Si (s, t) denotes the RBF RBF-lified signature kernel (Eq. 6).
Proof. Recall from Section 3,2 that we denote
x
of (Ix) E 2F and
Y!P = of E 1225.
where of is the random Fourier map defined in Eq. 4.
For fixed the R-CDE limit (Theorem 3.1) applied to the drivers XF and YF gives
1 ESN [ (I), Z (w))== = ks(s where
XXX Y
lim N N
kF
(24)
sig
Let 1: K RBF- By the PDE/Volterra characterization of the signature kernel (Theorem A.3), kr, [0, 2 + IR are the unique solutions of the two-parameter Volterra equations
kp(stt) = = [// qr(u, v) kr(u, v) du du,
and
k(s,t) 1 1.% q(u,u) ((u,))ku,, k(u, ((,,)) v) du du,
respectively, with driving kernels qF (u, v) == (X Yr y and q(u, v) = (Xu, Yo),, where H is the RKHS of the RBF feature map.
By Lemma 4, qF + q in L' ((0,7)3 almost surely (over the RFF draw), and there exists an envelope
1(t F yarl & C(w) | lelel. and |xx, Y) S C(w) | lidel
with I. ||اll|| E (' 7') and C(w) < co. The standard Volterra stability estimate via the two-parameter Grönwall inequality (Defranco, 1976), therefore yields
|kr klloc I exp(CCllãxlle lIúellz:) |gr alle.
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Consequently, (s, k(s,t) K2.9
Sig-RBF uniformly [)2 for p-almost every RFF draw. Taking F + in Eq. 24 and using this uniform convergence (together with the uniform bound implied by the envelope in the stability estimate) proves the theorem by dominated conver- gence.
B.2 PROOFS AND LEMMAS FOR SECTION 3.3
We first state and prove lemma ensuring that A' is well defined and convergent on group-like elements (signature increments). We then prove Lemma from the main text. Finally, we include short derivation of the smooth driver setting mentioned in Remark 3.2.
Lemma 6 (Absolute convergence of a signature increments). Let p 1, d E N, and let X E (RR with control w. Fix matrices AI...., Aa E End(R-) and set
R : max 15isd VN
|| <0.
For word in ...... define A N $ A4, ÀT the series
Ai and A by Eq. 11. Then for every (s,t)
Σ wEWa
Aw (Za,,
converges absolutely in End(R>), Consequently I'A (%(,)) is well-defined and (s,t) + (x) continuous.
By sub-multiplicativity and the definition of
|| N |4 ........ Aia N
[| S x* for |w/ k.
Group the series by word length and use the triangle inequality:
Σ || (Ko,t, w)l
(X,,,, w)
WEW
0 no
By Cauchy-Schwarz,
Σ 1K, S d= [ kk/=
where || I| is the Definition 2.1):
norm. Geometric p-rough paths satisfy the factorial decay (see
Op (p, r k +1)
|| (X..))l
for some Cp O that depends only on Hence
Σ || w)l S Cp Σ (2) w(s W.
+(
km0
By Stirling's formula, T(k/p + ~ (k/p) 4/pe thus the right-hand series converges for all w(s < oo. Absolute convergence implies the claim.
which outgrows any exponential;
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B.2.1 PROOF LEMMA
For convenience we restate Theorem and provide its proof.
Lemma. Let A be as in Eq. 11. If [0, + Re has bounded variation and ot := -ii A v.p
with A, E End((RA), ), then S (2) in Eq. 12 is the unique solution of the linear matrix CDE (x) = 5, (a) dx
(x) = Iddx E EEn((R
Proof. For x of bounded variation, the signature solves Chen's integral equation
- Sig(z)a. + ((a. Sig(z),,u dr..
with dr., (dr i.......... ... dr Applying the algebra homomorphism T (which sends concatenation @ to composition, and e; + A,) yields
IN-
L( (x) == T (S(i((())))) = IdN + (1.1)
S (1,((z) dr
dr = Σ A dr
i.e. (z) Sx) dxy with Sa) Idy ER Uniqueness follows from standard Picard iteration for linear matrix CDEs driven by bounded-varien variation paths.
B.2.2 REMARK 3.2 (SMOOTH-DRIVES DERIVATION)
Assume has bounded variation that X Sig(2). Starting from
2, = Zo + L (p(Zz))
expand dSS using S.A.A Ewews Au (Sig(())o,.
6: Zo + /'" (E Au d(Sig(())).) wi) (v((z))..
WeW
Write each non-empty word as and d(Sig(z)), 0) = 0 to get
@ (last letter i) and use d(Sig(z)o, wr)) (Sig(z)o,,: 1)) dx
Z, Zo Σ l
Σ
Aes (Sig(z)o, x) (+(()) dr'
WEW.
By multiplicativity of A (so Aga = AgAd),
Σ Aai (Sig(z)o, 3) = E
Aa (Sig(z)o,u @) k)) A. S Ar.
and hence
Z. = Zo + Σ 1"
d
(S A, ((z,)) dzi,
i=1 0
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B.3 PROOF OF THEOREM 3.4
For convenience we restate Theorem 3.4 and provide its proof.
Theorem. Let X E (IR)) and Y E GG2 (R4) be geometric rough paths. Let 2 (x) and Z, (?) be the solutions of Eq. 15 with = id and the same matrices {Ai) (with A₁ ~ Ex ii.d)).
Then for all 8, I [0,T]
lim N N BEN (Z, (2 (1)) = K K (s,
Z
where where Sig denotes the rough signature kernel defined in Eq. 7.
Proof. As in the smooth/CDE case, the (matrix) Dyson/Chen expansion under yp id gives
lim 1 EEN [1z (X), (x)) 100 N
lim EEN N00
N (AWw) Sig (X), (Y),
I.JEWa
where Wa introduced in Section 3.3 and (-) denotes the coordinate of the signature associated to the word (i.e. Sig (-) = (Sig(.),I).
From here, the argument mirrors Cass & Turner (2024), which also underpins the alternative proof of Theorem 3.1. In order to apply Lemma 5, we would like to exchange the limit and expectation and the double sum. By Fubini-Tonedl and the dominated convergence theorem, justify the exchange is enough show that
EEN
[|r (AW))
Sig
(25)
[L + +1
uniformly bounded in N. As the matrices A are Gaussian it holds that
N 1 EEN tr (AN)))
1
Een ||ApWlllop!
S k
[ 2
+1
N
for constant and where llop is the operator norm.
Then, by the factorial decay of (the signature of) rough paths (Definition 2.1), and the fact that the Le norm is at least as large as the L1 norm on /8n, there exists some >> 0 for which Eq. 25 is bounded by
+ +1) (Wx) m T("
VT(n + 1)T(m + 1)(m))
()T(" 7 +1)
+))[(
7,m==
where the inequality follows from logarithmic convexity r. By Stirling's formula, I(an+1 1) V2v (an)on+ --n as n + 00,
the numerator grows subfactorially relative to the product l'nn/p + 1)1 + 1), so the double series converges for any fixed (w, k). Hence, by an exchange of limits and an application of Lemma 5, we see that
lim 1 BeN [12 (X), N (3)) N - N
Σ
BeN [tr (AW.)) Sig 5X), (Y))
N
+1
|
Sig (x), Sig (Y),
(Xi Sig Y)lr(
K (s,t)
(.,(sttt
which concludes our proof.
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C ADDITIONAL EXPERIMENTAL RESULTS AND DETAILS
C.I EXPERIMENTAL SETUP
Preprocessing. We use the archive's pre-specified train/test splits. Each dataset is min-max scaled to 1,, 1}, and sequences are represented via piecewise- linear interpolation. We apply time and base- point augmentation, and tune the inclusion of lead- transform via grid search. All sequences resampled to length 200 following Toth et al. (2025).
Implementation. Random differential equation models are implemented in Jax in our library RandomS gJax. All the other benchmarks have been evaluated using the kSig library (Tóth et al., 2025) built on CuPy (Nishino Loomis, 2017). Both libraries use CuML to perform SVM/LinearSVM calculations on GPU.
Neural Baseline Configuration. NCDE and NRDE are implemented with an MLP vector field and a linear readout. Their hidden dimensions are chosen so that the total parameter count ap- proximately matches the feature budgets used for the random models, ensuring fair comparison. Training follows the standard protocol: Adam optimiser, early stopping on validation accuracy.
Compute. All experiments trained and evaluated times on each dataset, then the median test accuracy is taken.
single NVIDIA RTX 3090 GPU. Each approach is
Hyperparameter selection. As documented prior work on RFSF (Toth al., 2025), truncated signature kemels (Király & Oberhauser, 2019), and SigPDE (Salvi et al., 2021a), the influence of the scaling parameters introduced in Eq. 10 and in Section 3.3 is strongly data- and task-dependent, and there is currently principled procedure for selecting them priori. Consistent with this literature, we treat them as hyperparameters tuned via grid search.
Some general heuristics nevertheless can guide the design of the search ranges: higher truncation orders m are often beneficial for rougher paths (in the sense larger p-variation), while reasonable choices of a, on typically ensure that typical increments AX; remain of order at most one.
Hyperparameter grids.
We tune all models via grid search; the swept grids are listed below.
For all models using RFFs, the Fourier frequency scale is swept (0.01, 0.025, 0.05, 0.1, 0.25, 0.5, 1, 2.5, 5, 10, 25, 50, 100} x the bandwidth suggested by Toth al. (2025).
Model-specific ranges are:
RFSF: signature level M (2,3,4,5)
All Random Differential Equation models:
activation E {id, tanh, ReLU);
σ (0.1, 0.5, 0.75, 1.0, 1.25, 1.5; oB 0..1, 0.5}:
- oo E (0.0, 1.0, 1.5).
R RDE: signature level M E (2,3, 4, 5), capped so that the number of (oog))signature coordinates does not exceed the feature budget.
RF CDE: number of Fourier features F (32, 64, 128, 256, 512, 1024), capped at 50x the input dimension.
All feature-based models optionally apply feature normalisation (True/False).
Other baselines (SigPDE, RBF/GAK/RWS, SVMs) use standard grids over their key hyperparame- ters (kernel bandwidths, regularisation, warping parameters).
Remark C.1. SigPDE is evaluated only with an RBF base kernel (Salvi et al., 2021a), due to the superior empirical performance reported therein.
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C.2 ABLATION: NUMBER RANDOM FEATURES
We study sensitivity to feature count by repeating the main evaluation with N - 64 and N 500 random features. Tables and below mirror the main setting (training three runs per dataset and reporting mean accuracy).
Dataset
R-CDE
RF-CDE
R-RDE
RESF-DP
RFSF-TRP
NCDE
NRDE
Articulary WordRecog.
0.917
0.963
0.903
0.957
0.957
0.957
0.917
AtrialFibrillation
0,400
0.467
0.533
0.267
0.333
0.467
0.267
BasicMotions
1.000
1.000
0.975
0.975
1.000
0.975
0.975
Cricket
0.931
0.944
0.917
0.917
0.944
0.917
0.898
SigenWorms
0.420
0.611
0.594
0.701
0.755
0.675
0.420
Epilepsy
0.963
0.963
0.927
0.949
0.927
0.963
0,927
EihanolConcentration
0.308
0.338
0.361
0.430
0.418
0.385
0.361
FingerMovements
0.510
0,520
0.500
0.490
0.510
0.500
0,400
Handwriting
0,279
0.340
0.302
0.302
0,305
0,305
0.279
Libras
0.827
0.894
0.856
0.817
0.883
0.827
0.894
NATOPS
0.900
0,900
0.889
0.789
0,889
0.833
0.789
RacketSports
0.750
0.796
0,691
0.747
0.782
0.796
0.747
Sed/RegulationnSCPI
0.825
0.853
0.857
0.880
0.884
0.846
0.857
SellRegulationSCCP2
0.522
0.533
0.567
0.550
0.494
0.517
0.550
StandWalkJump
0.267
0.469
0,400
0.400
0,333
0.333
0,333
UWaveGestureLiibrary
0.834
0.822
0.897
0.825
0.803
0.789
0.822
Avg. acc. (t) Avg. rank (1)
0.666
0.713
0.698
0.687
0.701
0.692
0.652
4.437
2.500
4.094
4.156
3.375
4.186
5.250
Table 4: UEA test accuracies with N 64 random features. The neural baselines NCDE and NRDE comparable parameter budgets. Best result per in bold.
R-CDE
RF-CDE
R-RDE
RESF-DP
RFSF-TRP
SigPDE
Articulary WordRecognition
0.973
0.983
0.950
0.973
0.983
0.983
AtrialFibrillation
0.200
0.333
0.400
0,400
0.333
0.333
BasicMotions
1.000
1.000
0.975
1.000
1.000
Cricket
0.986
0.986
0.917
0.958
0.958
0.972
EigenWorms
0.458
0.664
0.612
0.824
0.817
0.794
Epilepsy
0.942
0.971
0.942
0.942
0.957
0.891
EdhanolConnenntration
0.319
0.407
0.375
0.517
0.414
0.460
FingerMovements
0.520
0.610
0.550
0.590
0.600
0.610
Handwriting
0.331
0,380
0.362
0,424
0.426
0,409
Libras
0.856
0.911
0.867
0.872
0.900
0.867
NATOPS
0.889
0.944
0.906
0.867
0.933
0.928
RacketSports
0.809
0.829
0.717
0.889
0.842
0.849
SellRegulationSCPP
0.877
0,881
0.843
0.894
0.881
0.904
SelfRegulationSSCP2
0.578
0.578
0.557
0.533
0.494
0.544
StandWalkJump
0.333
0.400
0.467
0.400
0.400
0.400
UWWeeGesttreLLibrayy
0.850
0.850
0.913
0.897
0.859
0.866
Avg. acc. (t) Avg. rank (1)
0.683
0.733
0.711
0.747
0.737
0.738
4.812
2.812
4.125
3.187
3.031
3.031
Table 5: UEA test accuracies with N = 500 random features. SigPDE is a kernel method (no random features), results are unaffected by N. For each row, the best result is highlighted in bold.
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Performance at N=64 Random Features Reducing the feature budget frrm 250 to 64 lowers curacy across all models, but the relative ordering changes noticeably. RF-CDE achieves both the best average accuracy and the best average rank among all methods (0.713, 2.50), and R- R-RDE also remains competitive with only a modest drop (0.708 + 0. (.698). In contrast, the signature-projection baselines exhibit larger declines (RESF-DP 0.726 0.687, RFSF-TRP 0.725 0.701). Easy datasets such as BasicMotions and Epilepsy stay saturated, while more complex datasets drive most of the differences across methods. Overall, the small-feature regime highlights that the random dif- ferential equation reservoirs retain good performance even when the feature budget is substantially constrained, whereas the neural baselines (NCDE, NRDE) struggle remain competitive this scale.
Performance at N=500 Random Features. The largest average gains appear for the randomized- signature baselines (RFSF-DP 0.726 - 0.747, RESF-TRP 0.725 - 0.737), while RF-CDE remains essentially flat (0.741 0.733) and R-RDE shows slight lift (0.708 0.711). with clearer im- Worms for RF-CDE: 0.630 - 0.664, UWaveGiesture
provements on harder sets (e.g., FigenWorms for R-RDE: 0.903 0.913). Easy tasks (e.g.. BasicMotions) are already saturated at near- near-perfect accuracy. In short, our random differential equation reservoirs already operate efficiently at 250 features; doubling N yields incremental benefits on subset of challenging datasets.
C.3 ADDITIONAL BASELINES FOR UEA CLASSIFICATION CLASSICAL KERNELS AND RFF
For completeness, we report standard time-series baselines; Random Fourier Features (RFF), RBF kernel SVM, Global Alignment Kernel (GAK), and Random Warping Series (RWS). RFF is eval- uated at two budgets (250 and 500 features), whereas RWS is reported only for the 250-feature setting. These baselines rarely achieve state-of-the-are performance on our suite serve as useful reference points.
RFF-250
RFF-500
RWS
GAK
RBF
Wordeleoyoon
0,980
0,980
0.970
0.977
0.977
AtrialFibrilltion
0.333
0.333
0.427
0.333
0.267
BasieMotions
0,925
0,925
0.995
1,000
0.975
Cricket
0.889
0.889
0.958
0.944
0.917
FogenWorms
0.431
0.431
0.578
0.511
0.496
Epilepsy
0.775
0.775
0.925
0.870
0.891
EthanolConcentration
0.316
0.316
0.284
0.361
0.346
FingerMovements
0.620
0.620
0.580
0.500
0.620
Handwriting
0.247
0.247
0.591
0.481
0.307
Libras
0.783
0.783
0.828
0.767
0,800
NATOPS
0.906
0.906
0.900
0.922
0.917
RacketSports
0.757
0.757
0.861
0.849
0.809
SeIiRegulationSCCI
0.894
0.894
0.829
0.915
0.898
SellRegulationSCP2
0.483
0,483
0.456
0.511
0,439
StandWalkJJmp
0.267
0.267
0.333
0.267
0.533
UWaveGestureLibbrayy
0.838
0.838
0.897
0.887
0.766
Avg. acc. (t)
0.679
0.679
0.721
0.703
0.706
Table 6: Baseline comparison on UEA datasets: Random Fourier Features (250/500), Random
Warping Series (RWS) number of features 250, Global Alignment Kernel (GAK), and RBF kernel SVM. Entries are test accuracies using the standard splits.
C.4 ROBUSTNESS TO MISSING OBSERVATIONS
We assess the stability of all models under partial observations using corrupted versions of the UEA multivariate sere-s datasets. Importantly, the corruption mechanism is applied only the test set, while training is always performed on the clean, unmodified training split. This isolates robustness at inference time.
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This experiment uses feature budget of N 64, matching the configuration examined in the ablation study, Accordingly, the numbers reported here are directly comparable to the N 64 results (Table 4) presented in Appendix C.2.
Corruption model. Given an input path X E RTxdd we generate a binary mask M E (0, 1 by independently removing entries with fixed probability p. We consider two regimes:
Low corruption: 20%,
High corruption: p 40%.
The corrupted path is defined coordinatewise as
missing,
Mr, = 0,
Xx,
Mr,i
Missing values are then imputed using simple linear interpolation along the time axis.
Evaluation protocol. All models are trained on the full, uncorrupted training set. During testing, the corrupicd-interoolate sequences are processed without modifying the training pipeline. Perfor- mance degradation as p increases provides direct measure of robustness to missing observations.
The hyperparameter grids and the general experimental setup follow exactly the Time-Series Clas- sification experiment and are reported in Section C.I.
R-CDE
RF-CDE
R-RDE
RFSF-DP
RESF-TRP
Articulary WordRecog.
0.896
0,953
0.903
0.953
0,943
AtrialFibrillatin
0.400
0,400
0.533
0.333
0.200
BasicMotions
0.975
1.000
0.975
0,900
1.000
Cricket
0.944
0.972
0.875
0.875
0.931
EigenWorms
0.458
0.594
0.594
0.611
0.664
Epilepsy
0.906
0.913
0.768
0.913
0.984
EdhanolComneentration
0.304
0.312
0.324
0.414
0.407
FingerMovements
0.510
0.520
0.510
0.510
0.520
Handwriting
0.279
0.302
0.279
0,300
0.228
Libras
0.867
0.906
0.733
0.806
0.878
NATOPS
0.844
0.867
0.844
0.778
0.867
RacketSports
0.743
0.763
0.539
0.612
0.697
SellRegulationSCPI
0.836
0,884
0.795
0.877
0.826
SelfRegulationSCCP2
0.533
0.505
0.550
0.483
0.483
StandWalkJump
0.267
0,400
0.333
0.267
0.200
UWaveGestureLLbbrryy
0.817
0.817
0.857
0.806
0.819
Avg. acc. (t)
0.661
0.694
0.650
0.652
0.665
Avg. rank (L)
3.438
1.938
3.406
3.406
2.812
Avg. acc. % decrease (4)
0.695
2.672
6.777
5.074
5.082
Table UEA test accuracies under 20% missing observations, where the corruption applied only to the test set. The number of features For each dataset, the best result is shown in bold.
Results. Across the 20% and 40% missing-value settings, all models naturally experience a drop in accuracy, but the magnitude of this degradation varies substantially. RF-CDE remains the strongest overall performer, retaining the best average accuracy and best average rank in both corrupted regimes, with only moderate decreases (-2.7% at 20% missing, -5.7% at 40%). R-CDE also degrades slowly (-0.7% and 5.9%%, remaining competitive despite its simpler architecture. By contrast, the RFSF baselines exhibit noticeably larger losses, particularly at higher corruption levels.
Unexpectedly. R-RDE does perform well in this experiment, despite the theoretical robustness of signature features to missing data. Its accuracy drops more sharply than for CDE-based reser- voirs. We suspect this partly due the short UEA time series and the relatively small chunk/step
28

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Published as a conference paper at ICLR 2026
size used in the log-ODE discretisation; in longer forecasting settings, with larger log- signature chunks, we expect the intrinsic stability of signature features to manifest more clearly.
At the per-dataset level, performance declines mostly monotone as corruption increases, with RF-CDE continuing to dominate on structured motion datasets, while R-RDE remains competitive few highly irregular tasks. Overall, the results suggest that CDE-based reservoirs com- paratively robust to missing inputs, while log-ODE RDEs may require longer temporal horizons or coarser signature blocks fully leverage their theoretical advantages.
The results are reported in Tables 7 and 8.
R-CDE
RF-CDE
R-RDE
RFSF-DP
RESF-TRP
Articulary WordRecog.
0.840
0.940
0.840
0.957
0.937
AtrialFibrillation
0.400
0.333
0.400
0.333
0.200
BasicMotions
0.775
0.975
0.925
0.725
0.800
Cricket
0.972
0.944
0.847
0.819
0.931
WigenWorms
0,458
0.489
0.489
0.489
0.542
Epilepsy
0.884
0.913
0.645
0.826
0.760
EthanolConcentration
0,285
0.304
0.319
0.418
0.414
FingerMovements
0,510
0.520
0.500
0.560
0.540
Handwriting
0.279
0.279
0.261
0,267
0.221
Libras
0.830
0.883
0.733
0.783
0.850
NATOPS
0,710
0,801
0.755
0.755
0.844
RacketSports
0,651
0.697
0.454
0.539
0.579
SelfRegulationSCPI
0.843
0.887
0.771
0.867
0.812
SellRegulationSCCP2
0.522
0.522
0.517
0.517
0.478
StandWalkJJmp
0.267
0.469
0.400
0,333
0.267
UWWveGestuureLibrryy
0.803
0.812
0.821
0.812
0.821
Avg. acc. (t)
0.627
0.673
0.604
0.625
0.624
Avg. rank (4)
3.281
2,000
3.594
3.062
3.062
Avg. acc. % decrease (4)
5.858
5.651
13.36
9.058
10.89
Table UEA test accuracies under 40% missing observations, where the corruption applied only to the test set. The number of features For each dataset, the best result is shown in bold.
C.5 HURST CLASSIFICATION TASK
Fractional Brownian Motion. Fractional Brownian motion (fBm) (B," he([. with Hurst expo-
nent H (0, 1) the unique mean-zero Gaussian process with covariance
E B B (12 2 - 2(2
(26)
The parameter H governs the roughness of sample paths:
H < yields negatively correlated increments and rough trajectories;
recovers standard Brownian motion;
Н > yields smoother trajectories with persistent increments.
For H < J. the expected p-variation is finite only for p > 1/H, changes in H induce clear geometric differences. Recovering H from sample paths is therefore a natural benchmark for models based on signature features, kernels, or controlled/rough differential equations.
Data Generation.
We generate univariate fractional Brownian motion using the standard Davies-
Harte method. For each class
H E (0.05, 0.15. ..., 0.7.......7).
produce:
29

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Published as a conference paper at ICLR 2026
Muria 50 samples per class,
25 samples per class,
length 256 time steps,
dimension d 3 by stacking three independent [Bm realisations with the same H.
Thus each input sample is tensor X NR and the label is the discrete index of the Hurst value.
VI and V2 Variants. The first variant (V1) uses the raw fBm trajectories. The second variant
(V2) applies a per-sample standardisation
X
N
N N E -
P4
X
of
(27)
N
This transformation removes global amplitude differences and forces the model to rely purely on geometric and temporal structure. Since fBm trajectories with different H can have substantially different variances, this normalisation creates a more challenging benchmark in which roughness must be inferred independently of scale.
Number of Features. All Random Differential Equation models and RFSF baselines are evaluated with feature budget of 64. The neural baselines (Neural CDE and Neural RDE) are configured to have a comparable number of parameters to ensure a fair comparison across architectures.
Benchmarks. In this experiment we compare our models exclusively with feature-based base- lines: Random Fourier Signature Features (RFSF) with diagonal and tensor projections, Neural Controlled Differential Equations (NCDE) (Kidger et al., 2020), and Neural Rough Differential Equations (NRDE) (Morrill et al., 2021). We additionally evaluated plain Random Fourier Features, which consistently failed to exceed 30% accuracy in any configuration and are therefore omitted for clarity.
Results.
The results are reported in Table 3 in Section 4.2.
30


Reviews:
[{"review_id": "vozEESV1oZ", "reviewer": "Reviewer_Fv7a", "summary": "The paper introduces random-feature-based continuous-time reservoir models (RF-CDEs and R-RDEs) for time-series tasks. Both approaches approximate continuous-time dynamics using random neural controlled differential equations (N-CDEs) as reservoirs, while training only a linear readout layer. In essence, this provides a mathematical/software realization of the physical reservoir computing (PRC) framework. The authors show that, in the infinite-width limit, these models converge to kernel methods: RF-CDEs to an RBF-lifted signature kernel, and R-RDEs to a rough signature kernel. This establishes a theoretical bridge between neural and kernelized path representations. Experiments on standard time-series benchmarks demonstrate competitive accuracy and high training efficiency.", "questions": "- How does the expressive capacity of RF-CDE and R-RDE compare to that of a learnable Neural CDE with an equivalent number of parameters? Is there any theoretical or empirical analysis addressing this? \n\n- The proposed models can be regarded as theoretical or software analogues of the PRC, which in principle requires only a linear output layer. However, practical hardware implementations of PRC often include a nonlinear output layer to achieve performance on par with NNs. Have the authors explored how introducing a small nonlinear readout layer would affect the performance of RF-CDE or R-RDE?\n\n - In the infinite-width limit, the RF-CDE and R-RDE converge to Gaussian processes with signature-kernel covariances. Could this be interpreted as placing a specific functional prior over neural CDE dynamics, analogous to the GP prior that emerges in the NTK or NNGP limits?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper bridges two compelling areas: reservoir computing and the infinite-width limit of neural CDEs. To me, this connection seems novel.\n-  The work is technically sound and has solid theoretical grounding. It proves asymptotic kernel equivalence and establishes existence and uniqueness of the limiting dynamics. \n- The authors provide a computational complexity analysis; the proposed linear-readout-only training offers efficiency gains.\n- Although the paper is primarily theoretical, it includes empirical benchmarks on standard time-series classification datasets. RF-CDE achieves the notable average performance among the baselines. \n- The manuscript is theory-heavy but well-organized.", "weaknesses": "I do not identify any critical drawbacks in this paper. However,\n\n- The experimental section focuses mainly on ablations of the proposed models and their variants. This improves completeness, but practitioners may be unsure about the broader practical advantages.\n\n- In particular, while the motivation is framed in terms of neural CDEs + PRC, the paper does not compare against neural CDEs or other end-to-end trained deep learning architectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9VTkW1tVKe", "reviewer": "Reviewer_d2Ug", "summary": "The paper proposes a training-efficient family of time-series models that combine random reservoirs with controlled/rough differential equations (CDE/RDE). Two main variants are introduced:\nRF-CDE: lift inputs with Random Fourier Features and evolve them through a random CDE; in the infinite-width limit the model induces the RBF-lifted signature kernel.\nR-RDE: operate directly on rough paths via a log-ODE discretization; in the infinite-width limit the model induces the rough signature kernel.\nOnly a linear readout is trained (reservoir computing spirit), yielding fast fitting and linear-in-samples scaling.", "questions": "1. May you provide non-asymptotic bounds?\n2. How do RF-CDE and R-RDE handle missing data, time jitter, or heavy noise? Any controlled experiments varying p-variation or noise levels?\n3. Are there stability issues for large m or rough drivers For R-RDE’s log-ODE?\n4. can you analyze why and provide guidance on when to choose R-RDE over RF-CDE?\n5. Any preliminary results on forecasting or imputation to bolster the practical case?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Clean theoretical framing with meaningful limits. RF-CDE -> RBF-lifted signature kernel and R-RDE -> rough signature kernel in infinite width; the results connect random differential equations to signature kernels in a principled way.\n2. Train only a linear head on top of fixed random dynamic, this is simple and fast to fit in practice\n3. Clear accounting for feature-extraction costs; random-feature models scale linearly in sequence length l.", "weaknesses": "1. The main theorems characterize infinite-width limits. There are no non-asymptotic approximation or generalization bounds to explain when a few hundred features suffice\n2. All evaluations are classification on UEA (16 datasets). No forecasting, imputation, irregular sampling, or robustness to noise/missingness. \n3. While RF-CDE averages best among random-feature models, R-RDE trails (avg. 0.708). Some difficult datasets (EigenWorms, Handwriting) show large gaps to SigPDE/RFSF.\n4. Ablations are thin. Beyond doubling features (250→500), there’s little exploration of RFF count F, signature truncation/order M, or log-ODE level m. \n5. There is no one figure in the whole paper, which will increase the difficulty for the reader to understand and capture the key designs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FqExZiqkr5", "reviewer": "Reviewer_tJMR", "summary": "The paper proposes random-feature reservoirs in continuous time built from controlled/rough differential equations, with only a linear readout trained. Two main variants are introduced: RF-CDE, which first lifts inputs with Random Fourier Features then evolves them through a random CDE; and R-RDE, which acts directly on geometric rough paths using a log-ODE discretization with log-signature inputs.", "questions": "- Do you have non-asymptotic approximation or error bounds that relate N, F, and signature truncation to excess risk or kernel approximation error?\n\n- How sensitive are RF-CDE and R-RDE to $\\sigma_A$, $\\sigma_b$, $\\sigma_0$, F, and log-signature truncation (m)? Please add systematic sweeps or ablation studies.\n\n- Why not include trained Neural CDE and modern sequence models as baselines for classification (such as variants of Neural CDEs or stable SDEs)?\n\n- Do these models maintain advantages on forecasting or irregular sampling tasks, where continuous-time structure should help? Have you considered classification with missing observations or features? \n\n- Can you report wall-clock time and peak memory across UEA for fixed accuracy targets, to validate Table 1's asymptotics?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The writing is high-quality and mathematically literate. The background sections on rough paths, signatures, and kernels are compact, accurate, and helpful for positioning the work.\n\n- The paper cleanly recalls the R-CDE limit to the signature kernel and extends it with two variants whose limits are the RBF-lifted signature kernel (RF-CDE) and the rough signature kernel (R-RDE). The statements (Theorems 3.2 and 3.4) are explicit.\n\n- The paper correctly frames the models as training-efficient reservoirs with a linear readout. It provides a clear asymptotic cost analysis that contrasts the linear-in-length feature extraction with the quadratic scaling of kernel Gram matrix approaches.", "weaknesses": "- The main theorems are in the infinite-width setting. The paper does not provide non-asymptotic approximation rates or generalization/error bounds for practical feature counts. \n\n- Only UEA classification is considered. There are no forecasting or irregular sampling experiments, despite the continuous-time claim. The asymptotic table is helpful, but there are no runtime or memory measurements on the UEA suite to validate the linear-in-length advantage or to quantify the cubic term in R-RDE.\n\n- The algebraic mapping ($\\Gamma_A$ and $\\Pi_B$) definitions are dense. It is not clear how the chosen truncation level (m) and Hall/Lyndon basis size affect stability, feature variance, and compute in practice, or how these are tuned.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xnKZSPvH2Q", "reviewer": "Reviewer_vGCM", "summary": "In this paper, the authors introduce a training-efficient framework for time-series learning by combining random features with CDEs. It uses randomly parameterized CDEs as continuous-time reservoirs, requiring training only a linear readout layer. Two variants are proposed: RF-CDEs, which lift inputs via random Fourier features, and RRDEs, which operate on rough-path inputs using log-signatures. In the infinite-width limit, these methods recover the RBF-lifted and rough signature kernels, offering a unified perspective on reservoir computing and path-signature theory. In addition to strong theoretical superiority, experiments on UEA datasets demonstrate that the proposed models achieve competitive or state-of-the-art performance.", "questions": "1.\tIn the experimental part of paper “Random Fourier Signature Features”, the truncated Signature Kernel exhibits a strong performance, looks even better than the newly proposed two variants, could the authors clarify this?\n2.\tIn line 240, should the term “homogeneous” be corrected to “inhomogeneous”? \n\n\nTypos:\n\n1. Line 122: Remove the word “lift” before $x$\n\n2. In Eq (4), $\\omega_M$ should be $\\omega_F$", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- The idea of bridging the random CDE with reservoir computing is both novel and interesting. Under this general idea, the two proposed variants offer distinct and independent advantages.\n- This paper presents solid theatrical contribution. the authors prove kernel convergence in the infinite-width limit, and the finite-dimensional computation remains efficient and scalable. \n- The paper is well-organized and clearly written, making the technical details accessible and the main ideas easy to follow.", "weaknesses": "- The connection between the proposed methods with reservoir computing should be strengthened.\n- The datasets in UEA benchmarks are relatively small. Evaluating the proposed methods on larger-scale benchmarks such as the Long Range Arena (LRA) would strengthen the empirical claims and better demonstrate efficiency.\n- Some baselines are lacking, a comparison with learning-based path signature approaches would be beneficial, such as [1] [2]\n\n[1] Morrill, James, et al. \"Neural rough differential equations for long time series.\" ICML, 2021.\n\n[2] Walker, Benjamin, et al. \"Log neural controlled differential equations: The lie brackets make a difference.\" ICML, 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)