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linear_esn

A reservoir with the nonlinearity removed and no readout attached: the model's output is the state sequence of a linear dynamical system. This is an analysis instrument, not a forecaster.

Wiring

Input → Reservoir(activation="identity")

With identity activation the update reduces to a driven linear recurrence,

\[x_t = (1-a)\,x_{t-1} + a\,\bigl(W x_{t-1} + W_{\text{fb}}\,u_t + b\bigr),\]

so the eigenvalues of \(W\) determine the dynamics: spectral_radius and the topology spec act on the linear spectrum directly, with no saturation involved. The factory forces activation="identity" and accepts no readout arguments; passing readout_alpha or output_size raises TypeError.

wiring

Model graph: the input drives a single identity-activation reservoir whose states are the output.

Use

import torch
from resdag.models import linear_esn

model = linear_esn(reservoir_size=200, feedback_size=1)
states = model(torch.randn(1, 500, 1))    # (1, 500, 200) — states are the output

# Effective dimensionality of the linear response
S = torch.linalg.svdvals(states[0])
participation = (S.sum() ** 2 / (S**2).sum()).item()

Use cases:

  • Spectral analysis. Measure the effect of a topology and spectral radius on the state dynamics without tanh saturation: compare the state spectrum across topology specs with the nonlinearity removed.
  • Memory-capacity studies. Linear reservoirs are the reference case in short-term-memory analysis; drive with white noise and regress delayed inputs from states to trace the capacity curve.
  • Ablation baseline. Any gain a nonlinear ESN shows over this model on the same topology is attributable to the nonlinearity, not the recurrence.

Parameters

Parameter Default Notes
reservoir_size, feedback_size required units, input dim — no output_size
topology, feedback_initializer None any initialization spec
spectral_radius 0.9 acts directly on the linear spectrum
leak_rate 1.0 1.0 = no leak
bias, trainable True, False random bias on; frozen weights
**reservoir_kwargs — forwarded to ESNLayer; activation is fixed to "identity"

Reference

H. Jaeger, Short term memory in echo state networks, GMD Report 152, German National Research Center for Information Technology (2001) — the canonical memory-capacity analysis for linear reservoirs.

See also

  • headless_esn — the same readout-free pattern with the nonlinearity kept.
  • Layers — ESNLayer, the single component this factory wraps.
  • Models reference — full factory signature.