dimensionality 0.1.3

Bias-corrected participation ratio estimator for measuring dimensionality of neural representations

Sample-size invariant measure of dimensionality

Bias-corrected participation ratio (PR) estimators for measuring the dimensionality of neural representation manifolds, as introduced in:

Estimating Dimensionality of Neural Representations from Finite Samples Chanwoo Chun*, Abdulkadir Canatar*, SueYeon Chung, Daniel Lee ICLR 2026

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📐 Background

Given a neural activation matrix $\Phi \in \mathbb{R}^{P\times Q}$ (P stimuli × Q neurons), the participation ratio

$$ \gamma =\frac{\left(\sum_i \lambda_i \right)^2}{\sum_i \lambda_i^2} $$

is a soft count of the number of nonzero eigenvalues of the stimulus covariance $K=\frac{1}{Q}\Phi\Phi^\top$. The naive estimator is severely biased downward when P or Q is small — it behaves approximately as a harmonic mean of P, Q, and the true $\gamma$:

$$ \mathbb{E}\left[ \frac{1}{\gamma_{\text{naive}}}\right] \approx \frac{1}{P}+ \frac{1}{Q} +\frac{1}{\gamma} $$

This package provides unbiased estimators that correct for finite P and/or Q by averaging over disjoint index sets.

Estimator	Corrects	Use when
γ_naive	nothing	baseline reference
γ_row	row (stimulus) sampling bias	full neuron access, subsampled stimuli
γ_col	column (neuron) sampling bias	full stimulus access, subsampled neurons
γ_both	both row and column bias	subsampled neurons, subsampled stimuli

An additional participation_ratio_finite estimator handles the case where Φ is a submatrix sampled without replacement from a large-but-finite R×C matrix (Appendix A.6 of the paper).

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📦 Installation

pip install dimensionality

Or, for development:

git clone https://github.com/badooki/dimensionality.git
cd dimensionality
pip install -e ".[dev]"

Dependencies: numpy >= 1.24, opt_einsum >= 3.3. Python ≥ 3.9.

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🚀 Quick start

import numpy as np
from dimensionality import participation_ratio

# Phi: P stimuli × Q neurons  (do NOT pre-center)
Phi = np.random.randn(200, 100)

# Default: γ_both (bias-corrected for both row and column subsampling)
gamma = participation_ratio(Phi)
print(gamma)

All four estimators

result = participation_ratio(Phi, return_all=True)
# result['naive'], result['row'], result['col'], result['both']

Two-trial noise correction

When two independent repeat trials are available for the same stimuli and neurons, the cross-trial construction removes additive and multiplicative noise bias:

gamma = participation_ratio(Phi1, Phi2)

Return numerator and denominator separately

result = participation_ratio(Phi, return_parts=True)
# result['both'], result['A'], result['B']

# Combined with return_all:
result = participation_ratio(Phi, return_all=True, return_parts=True)
# result['naive'], result['A_naive'], result['B_naive'], ...

Neuron dimensionality

To estimate dimensionality along the neuron axis (centering across stimuli), transpose the matrix:

gamma_neuron = participation_ratio(Phi.T)

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🔢 Finite underlying matrix

When Φ is a P×Q submatrix sampled without replacement from a finite R×C population matrix, use participation_ratio_finite:

from dimensionality import participation_ratio_finite

gamma = participation_ratio_finite(Phi, R=5000, C=2000)

# With noise correction:
gamma = participation_ratio_finite(Phi1, R=5000, C=2000, Phi2=Phi2)

# Also return the naive estimate:
result = participation_ratio_finite(Phi, R=5000, C=2000, return_naive=True)
# result['gamma'], result['naive']

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📊 Subsampling sweep

To assess how the estimate converges with sample size, sweep over P or Q:

from dimensionality import sweep_dimensionality, plot_sweep

# Sweep over number of stimuli; keep all neurons
result = sweep_dimensionality(Phi, axis='P', n_trials=20)

# result['values']  — array of P values used
# result['naive'], result['row'], result['col'], result['both']  — mean estimates
# result['both_sem']  — standard error of the mean

fig, ax = plot_sweep(result, true_d=50)

To sweep over number of neurons instead:

result = sweep_dimensionality(Phi, axis='Q')

For the finite estimator:

result = sweep_dimensionality(Phi, axis='P', estimator='finite', R=5000, C=2000)
# result['naive'], result['gamma']

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⚠️ Important: do not pre-center

The bias corrections rely on an algebraic three-term centering structure built into the estimator formulas. Subtracting column means from Φ before passing it to the estimator introduces statistical dependencies between rows that break the bias correction. Pass the raw activation matrix directly.

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🔧 API reference

participation_ratio(Phi, Phi2=None, *, return_all=False, return_parts=False)

Estimate the task dimensionality (PR of the centered covariance) of Φ.

• Phi — raw activation matrix, shape (P, Q); P ≥ 4, Q ≥ 2.
• Phi2 — optional second trial for noise correction.
• return_all — if True, return dict with all four estimator variants.
• return_parts — if True, include numerator A and denominator B.

Returns a scalar (γ_both) by default, or a dict when either flag is set.

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participation_ratio_finite(Phi, R, C, Phi2=None, *, return_naive=False, return_parts=False)

Estimate the PR of the full R×C matrix from the observed P×Q submatrix.

• R, C — number of rows/columns in the full underlying matrix; R ≥ P, C ≥ Q.
• return_naive — if True, also return the (uncorrected) naive estimate.
• return_parts — if True, include numerator A and denominator B.

Returns a scalar by default, or a dict when either flag is set.

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sweep_dimensionality(Phi, axis='P', values=None, n_trials=20, Phi2=None, estimator='infinite', R=None, C=None, ...)

Run a subsampling sweep. Returns a dict with mean estimates and SEMs at each value. See docstring for full parameter list.

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plot_sweep(result, ax=None, true_d=None, title=None, figsize=(5, 4))

Plot the output of sweep_dimensionality. Returns (fig, ax).

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🗂️ Repository structure

src/
  dimensionality/
    __init__.py        # public API
    _core.py           # quartic einsum helper
    estimators.py      # participation_ratio
    finite.py          # participation_ratio_finite
    sweep.py           # sweep_dimensionality
    plot.py            # plot_sweep
tests/
  test_estimators.py
examples/
  demo.ipynb           # interactive walkthrough on synthetic data
  synthetic.py         # Figure 1 reproduction

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📄 Citation

If you use this package, please cite:

@inproceedings{chun2026estimating,
  title     = {Estimating Dimensionality of Neural Representations from Finite Samples},
  author    = {Chun, Chanwoo and Canatar, Abdulkadir and Chung, SueYeon and Lee, Daniel},
  booktitle = {International Conference on Learning Representations},
  year      = {2026},
}

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License

MIT