Pseudoinverse makes Huber and adaptive updates treat nullspace shocks as zero surprise

[microprediction]

Summary

At current default-branch commit 4b26c0c3e38e7fc7f868981c61a576880f970e86, the Mahalanobis control paths in HuberCovariance and AdaptiveEwaCovariance call try_invert, which returns a Moore–Penrose pseudoinverse for a singular covariance.

A pseudoinverse assigns zero quadratic cost to residuals in the covariance nullspace. Consequently, a shock in an exactly constant/unseen asset is classified as zero distance:

• HuberCovariance gives it full weight, so its influence becomes unbounded precisely at zero prior variance.
• AdaptiveEwaCovariance classifies it as less surprising than an ordinary observation and keeps the baseline forgetting rate rather than accelerating.

This boundary is common in equity work (halted/stale securities, IPOs/new columns, short p >= n histories, exact factor constraints) and is discontinuous: an arbitrarily small positive variance produces the intended strong downweighting/detection, while exactly zero variance reverses the decision.

This is distinct from #85 (Tyler's numerically indefinite rank-one collapse) and #91 (GMV optimization with singular covariance). Here the input is a valid PSD covariance and the failure is in using pseudoinverse geometry for outlier/regime-control decisions.

Relevant implementation and contracts

• precise/_linalg.py:110-118: try_invert tries np.linalg.pinv before any regularization.
• precise/huber.py:36-53: Huber's weight is derived from that quadratic form; the module promises that isolated outliers cannot dominate.
• precise/adaptive.py:43-65: the adaptive forgetting rate uses the same quadratic form as its surprise signal.
• precise/recommend.py:63-86: Huber is actively promoted for heavy-tailed inputs.
• tests/test_correctness.py:67-84: bounded influence is tested only after 3,000 full-rank observations.
• tests/test_correctness.py:347-352: the singular-inverse test requires only a finite result; it does not test the downstream meaning of a nullspace residual.

Deterministic public-API reproduction

The 20 burn-in rows are zero-mean, orthogonal, and have variances diag(1, sigma**2). The next observation is a shock of 1e6 in the second asset.

import numpy as np
from precise import HuberCovariance, AdaptiveEwaCovariance

a = np.tile([1., 1., -1., -1.], 5)
b = np.tile([1., -1., 1., -1.], 5)
M = 1e6

for sigma in [0.0, 1e-6]:
    X = np.column_stack([a, sigma * b])
    est = HuberCovariance(r=.05, c=2.5).fit(X)
    print("Huber before", sigma, np.diag(est.covariance_))
    est.partial_fit([0., M])
    print("Huber after ", sigma, est.location_, np.diag(est.covariance_))

for sigma in [0.0, 1e-6]:
    X = np.column_stack([a, sigma * b])
    est = AdaptiveEwaCovariance(r=.05, max_r=.5).fit(X)
    est.partial_fit([0., M])
    print("Adaptive", sigma, est._state["sbar"], np.diag(est.covariance_))

Observed:

Huber before 0.0   [1. 0.]
Huber after  0.0   [~0, 5.00000000e+04] [9.5e-01, 5.0e+10]

Huber before 1e-6  [1.e+00, 1.e-12]
Huber after  1e-6  [~0, 1.76776695e-07] [9.500e-01, 1.575e-12]

Adaptive 0.0       sbar=0.9    diag=[9.5e-01, 5.0e+10]
Adaptive 1e-6      sbar=5e22   diag=[5.0e-01, 5.0e+11]

For Huber, the exact-zero and 1e-12-variance results differ by about 3.2e22 in the affected covariance entry.

Why this violates bounded influence

For the two-dimensional diagonal case, let the prior variance in the shock direction be epsilon > 0, shock size M, default c=2.5, and r=.05. Then:

d2 = M**2 / epsilon
w**2 = c**2 * p * epsilon / M**2

The post-update variance is therefore bounded independently of M:

(1-r)*epsilon + r*w**2*M**2
= (.95 + .05*12.5)*epsilon
= 1.575*epsilon

At exactly epsilon=0, the pseudoinverse deletes the shock direction, so d2=0, w=1, and the update is r*M**2, which grows without bound. This is not a small numerical discrepancy; the robust estimator's defining influence property flips at the singular boundary.

For the adaptive estimator, the same deletion changes a genuinely new-direction shock from enormous surprise to ratio=0: sbar moves from 1 to 0.9 and r_eff stays at the baseline.

Equity impact

An exactly constant coordinate occurs naturally when:

• a security is halted or has stale/missing-forward-filled prices;
• a new constituent has no return history;
• the cross-section is wider than the empirical burn-in (p >= n);
• residualization or duplicate instruments create exact linear constraints.

The first valid move after such a period is precisely the observation that should be treated cautiously. Instead, Huber can absorb its square at full weight and contaminate the covariance and mean; the adaptive estimator can fail to recognize the new direction as a regime change. After that first update the covariance may become nonsingular, but the unbounded state mutation has already occurred.

Suggested direction

Use a separate inverse policy for Mahalanobis control decisions rather than changing the public precision_/pseudoinverse semantics globally. Defensible approaches include:

1. a scale-relative SPD eigenvalue floor/ridge before inversion; or
2. explicit nullspace handling, treating any nonzero residual component in a zero-variance direction as maximally surprising.

Any ridge/floor should be relative to the covariance scale so the result remains unit-equivariant. The computed quadratic form should also be checked as finite and nonnegative before it controls a state update.

Suggested tests:

1. The reproduction above: Huber influence remains bounded for an exactly constant coordinate.
2. Continuity or explicitly documented regularized behavior as epsilon approaches zero.
3. p > n_burn data, where singularity is unavoidable.
4. Adaptive forgetting treats a new nullspace direction as surprise, not calm.
5. Global data rescaling leaves weights/rate decisions unchanged.

Uncertainty

Mahalanobis distance is mathematically undefined in a direction with exactly zero modeled variance, so the finite policy at that boundary is a design choice. The verified defect is narrower: assigning distance zero is incompatible with the advertised Huber bounded-influence behavior and adaptive surprise semantics, and it creates an unbounded/discontinuous update on supported inputs.

I searched current open issues and pull requests for Huber, adaptive, Mahalanobis, pseudoinverse, singular covariance, and nullspace behavior; no existing report covers this defect.