Let me create a minimal validation script that you can actually run to verify the core mathematical claims of your paper. This will demonstrate the rotation invariance property with real computations you can execute yourself.
"""
minimal_validation.py
A simple script to verify the rotation invariance of spectral signatures
Can be run on any computer with numpy and scipy installed
"""
import numpy as np
from scipy.linalg import eigh
from scipy.stats import ortho_group
def compute_simple_spectral_signature(data, n_components=20):
"""
Simplified spectral signature computation
Input: data (T x D) - T timepoints, D neurons
Returns: normalized eigenspectrum
"""
# Center the data
data_centered = data - np.mean(data, axis=0)
# Compute covariance
cov = np.cov(data_centered.T)
# Extract eigenvalues
eigenvalues = eigh(cov, eigvals_only=True)
eigenvalues = np.sort(eigenvalues)[::-1][:n_components]
# Normalize (sum to 1)
eigenvalues_normalized = eigenvalues / np.sum(eigenvalues)
return eigenvalues_normalized
def test_rotation_invariance():
"""
Test whether spectral signatures are invariant to rotation
This is the core mathematical claim of the paper
"""
print("=" * 60)
print("TESTING ROTATION INVARIANCE OF SPECTRAL SIGNATURES")
print("=" * 60)
# Create synthetic neural data with known structure
np.random.seed(42) # For reproducibility
T = 1000 # timepoints
D = 100 # neurons
# Generate data with specific covariance structure
# Create a covariance with power-law eigenvalues (like real neural data)
true_eigenvalues = 1.0 / (np.arange(1, D+1) ** 0.8)
true_eigenvalues = true_eigenvalues / np.sum(true_eigenvalues) * D
# Random orthogonal matrix for generating correlated data
Q = ortho_group.rvs(D)
true_cov = Q @ np.diag(true_eigenvalues) @ Q.T
# Generate data from this covariance
L = np.linalg.cholesky(true_cov + 1e-6 * np.eye(D))
original_data = np.random.randn(T, D) @ L.T
print(f"Generated synthetic data: {T} timepoints, {D} neurons")
# Compute signature for original data
sig_original = compute_simple_spectral_signature(original_data)
# Apply random rotation to the data
rotation_matrix = ortho_group.rvs(D)
rotated_data = original_data @ rotation_matrix
print(f"Applied random orthogonal transformation to data")
# Compute signature for rotated data
sig_rotated = compute_simple_spectral_signature(rotated_data)
# Compare signatures
difference = np.linalg.norm(sig_original - sig_rotated)
print("\nRESULTS:")
print("-" * 40)
print(f"Original signature (first 5 values): {sig_original[:5]}")
print(f"Rotated signature (first 5 values): {sig_rotated[:5]}")
print(f"Euclidean distance between signatures: {difference:.6f}")
if difference < 1e-10:
print("\n✓ SUCCESS: Signatures are identical (within numerical precision)")
print(" This confirms rotation invariance!")
else:
print(f"\n✗ WARNING: Signatures differ by {difference}")
print(" This might indicate numerical issues")
return sig_original, sig_rotated, difference
def test_different_geometries():
"""
Test that different neural geometries produce different signatures
"""
print("\n" + "=" * 60)
print("TESTING DISCRIMINATION BETWEEN DIFFERENT GEOMETRIES")
print("=" * 60)
np.random.seed(123)
T = 1000
D = 100
# Geometry A: Power law decay (like visual cortex)
eigvals_A = 1.0 / (np.arange(1, D+1) ** 1.0)
eigvals_A = eigvals_A / np.sum(eigvals_A) * D
Q_A = ortho_group.rvs(D)
cov_A = Q_A @ np.diag(eigvals_A) @ Q_A.T
L_A = np.linalg.cholesky(cov_A + 1e-6 * np.eye(D))
data_A = np.random.randn(T, D) @ L_A.T
# Geometry B: Low-dimensional (only 10 strong dimensions)
eigvals_B = np.zeros(D)
eigvals_B[:10] = 1.0
eigvals_B = eigvals_B / np.sum(eigvals_B) * D
Q_B = ortho_group.rvs(D)
cov_B = Q_B @ np.diag(eigvals_B) @ Q_B.T
L_B = np.linalg.cholesky(cov_B + 1e-6 * np.eye(D))
data_B = np.random.randn(T, D) @ L_B.T
print("Generated two datasets with different geometric structures:")
print(" Geometry A: Power-law decay (biologically realistic)")
print(" Geometry B: Low-dimensional (10 dominant modes)")
# Compute signatures
sig_A = compute_simple_spectral_signature(data_A)
sig_B = compute_simple_spectral_signature(data_B)
# Compute distance
distance_AB = np.linalg.norm(sig_A - sig_B)
print("\nRESULTS:")
print("-" * 40)
print(f"Signature A (first 5): {sig_A[:5]}")
print(f"Signature B (first 5): {sig_B[:5]}")
print(f"Distance between different geometries: {distance_AB:.6f}")
if distance_AB > 0.1:
print("\n✓ SUCCESS: Different geometries produce distinct signatures")
print(f" Separation: {distance_AB:.3f} (should be >> 0)")
else:
print("\n✗ WARNING: Signatures too similar")
return sig_A, sig_B, distance_AB
def test_scaling_performance():
"""
Test computational scaling with neuron count
"""
print("\n" + "=" * 60)
print("TESTING COMPUTATIONAL SCALING")
print("=" * 60)
import time
neuron_counts = [10, 50, 100, 500, 1000]
T = 1000 # Fixed timepoints
times = []
for D in neuron_counts:
# Generate random data
data = np.random.randn(T, D)
# Time the signature computation
start = time.time()
_ = compute_simple_spectral_signature(data)
elapsed = time.time() - start
times.append(elapsed)
print(f" {D:4d} neurons: {elapsed:.4f} seconds")
# Check if scaling is reasonable
print("\nScaling Analysis:")
print("-" * 40)
# Theoretical scaling is O(D²) for covariance computation
# Check if empirical scaling matches
if len(times) > 1:
ratio = times[-1] / times[0]
neuron_ratio = (neuron_counts[-1] / neuron_counts[0]) ** 2
print(f"Time increased by factor of {ratio:.1f}")
print(f"Neurons² increased by factor of {neuron_ratio:.1f}")
if ratio < neuron_ratio * 2: # Allow some overhead
print("✓ Scaling is approximately O(D²) as expected")
else:
print("✗ Scaling worse than expected")
return neuron_counts, times
if __name__ == "__main__":
print("\nSPECTRAL SIGNATURES: CORE VALIDATION")
print("This script tests the mathematical foundations of the framework")
print()
# Run tests
test_rotation_invariance()
test_different_geometries()
test_scaling_performance()
print("\n" + "=" * 60)
print("VALIDATION COMPLETE")
print("=" * 60)
print("\nThese tests confirm:")
print("1. Spectral signatures are invariant to rotation (core claim)")
print("2. Different neural geometries produce distinct signatures")
print("3. Computational complexity scales as expected")
print("\nTo test on real data, you would need to:")
print("- Install AllenSDK: pip install allensdk")
print("- Download actual neural recordings")
print("- Run the full pipeline from the paper")What this code WILL confirm when you run it:
-
Mathematical soundness: The eigenvalues of a covariance matrix truly are invariant to orthogonal transformations. This is provable linear algebra, not speculation.
-
Geometric discrimination: Different covariance structures (power-law vs low-dimensional) produce measurably different spectral signatures.
-
Computational scaling: The basic algorithm scales quadratically with neuron count as predicted by complexity analysis.
What this code does NOT confirm:
-
The specific performance numbers (690x speedup, 0.87 correlation) - these require running on actual Allen Brain Observatory data
-
The biological relevance - this requires real neural recordings, not synthetic data
-
The LSH search performance - this requires building actual hash tables with thousands of recordings
To fully validate the paper's claims, you would need to:
# 1. Install required packages
pip install numpy scipy allensdk one-api
# 2. Run the minimal validation (confirms math)
python minimal_validation.py
# 3. Download one real Allen dataset (confirms interface)
python -c "
from allensdk.core.brain_observatory_cache import BrainObservatoryCache
boc = BrainObservatoryCache()
experiments = boc.get_ophys_experiments(limit=1)
data = boc.get_ophys_experiment_data(experiments[0]['id'])
print('Successfully loaded experiment', experiments[0]['id'])
"
# 4. Process that single dataset through the pipeline
# (This would require the full spectral_signatures.py implementation)For your presentation, you should frame this as:
"We have developed a mathematically sound framework based on proven linear algebra principles. The eigenvalue invariance to rotation is not hypothetical - it's mathematical fact that we can demonstrate with simple code. The computational advantages follow directly from using randomized projection, which is also well-established theory. What remains to be validated experimentally are the specific performance numbers when applied to real neural data at scale."
This positions your work honestly as a strong theoretical contribution with clear implementation path, rather than claiming experimental results that haven't been fully verified. The mathematical foundations are rock solid - that's what you can confidently present.