Quasi-Orthogonal Tensor-Train Reservoirs: Hyperspherical Reservoir Computing Beyond the Memory Wall

The scaling of reservoir computing is limited by the O(N²) memory and compute cost of the connectivity matrix W. We combine two ingredients whose compatibility is, to our knowledge, previously unexploited: (i) hyperspherical reservoirs, whose state update x = z/‖z‖ uses the sphere normalization as its only nonlinearity — the only practical nonlinearity that never increases tensor-train (TT) rank — and (ii) TT-structured connectivity W(β) = β·Q_kron + (1−β)·W_tt, where the Kronecker-product orthogonal backbone Q_kron is an exactly orthogonal N×N operator of TT rank 1. The mixed connectivity, stored in at most rank 1+χ_w, matches dense-W task performance where dense is feasible (315× compression at N = 1024) and runs unchanged at compression factors of 7.3×10⁶ where a dense W cannot exist (N = 262,144: dense W 512 GB, TT form 73 KB), and linear memory capacity deepens with N at essentially fixed parameter cost: a delay-400 reconstruction task fails at N = 1024 (r² = 0.11) and succeeds at N ≥ 16,384 (r² = 0.92). On the theory side we show that normalized reservoirs are constitutively critical — their largest tangent-space Lyapunov exponent is ≈ 1 for any parameters, so standard stability metrics carry no information — and identify the mean one-step growth rate ḡ along the actual error direction as the quantity that decides the fate of state-truncation errors. We derive and validate the amplification law A ≈ min{(1−ḡ²)^(−1/2), √2/δ̄} on two independent measurement grids (96% of 45 and 94% of 52 configurations predicted within a factor of two). The convex mixing parameter β exhibits a non-interference dip of the normalization denominator, ⟨‖z‖⟩ ≈ √(β² + (1−β)²), which explains a runaway zone at intermediate β (amplification up to 34×) and a quasi-isometric stability band β ∈ [0.8, 0.85] where memory capacity is near-maximal while truncation-error amplification stays bounded (≤ 4 across seeds). All experiments run on a plain CPU; the reference implementation (sphertt, pure NumPy) is released as open source at https://github.com/Volta-0936/sphertt (archived: DOI 10.5281/zenodo.21433254).