Entroptics

Find the real structure in noisy multichannel data — how many components it holds, what they are, and how fast they decay and oscillate — without choosing a rank or tuning a threshold.

Give Entroptics any 2-D array: time × sensors, a spectrogram, a radio waterfall, a stack of embeddings. It tells you how many independent components stand above the noise, separates them from the noise exactly, and reads each one's frequency and decay rate. The only setting is the false-alarm rate you are willing to accept (5% by default).

Install

pip install entroptics              # numpy only
pip install "entroptics[torch]"     # + torch tensors and the GPU

Python ≥ 3.10 and numpy ≥ 2.0. From a checkout, pip install -e ".[dev]" adds the test and figure dependencies.

Why use it

• No knobs. The number of components is decided against the level pure noise of the same shape would reach, at the false-alarm rate you set.
• It finds the right count. On planted components above the noise, it recovers the true number in all 36 cases tested, where Gavish–Donoho, AIC and MDL get 89–95% right (validation, experiment 17).
• Calibrated on noise, any noise. On pure noise it reports a component at the rate you ask for, whatever the noise's distribution: Gaussian, Student-t, skewed, count and very heavy-tailed noise, at equal or unequal channel levels, with cells missing (benchmarks).
• Lossless. The signal it keeps plus the residual it returns is your input, to floating-point round-off.
• Frequency and decay together. It reads how fast each component dies away as well as its frequency, between the FFT's bins, across gaps in the record, and next to a much stronger tone (vs the FFT).
• Messy data welcome. Missing cells, dead or stuck channels, channels at very different levels, few samples across many channels.
• Light and fast. numpy only; torch (and a GPU) when you pass a tensor. Streams at O(F²) per frame, with state you can save, resume and splice. On multichannel records it is faster than the FFT at the same job: with 16 channels and 16384 samples or more, its count, frequencies and decay rates take 1.8–2.7× less time than an FFT pipeline that computes them, and from 262144 samples about what the bare FFT alone takes. On a single channel the FFT pipeline is faster (cost).

When to reach for it

Examples

How many components, and the signal apart from the noise

import numpy as np
from entroptics import Aperture

rng = np.random.default_rng(1)
signal = rng.standard_normal((600, 3)) @ rng.standard_normal((3, 40))   # 3 sources over 40 sensors
W = signal + rng.standard_normal((600, 40))                             # plus noise

ap = Aperture(W)
ap.projection().K_signal                    # 3
Aperture(rng.standard_normal((600, 40))).projection().K_signal         # 0 on pure noise

clean, info = ap.extract()                  # clean + info["residual"] == W, to 4e-16
# distance from the true signal: 0.562 for the raw record, 0.184 for clean

Frequencies and decay rates of a single signal

from entroptics import koopman_lift

t = np.arange(1024)
y = np.exp(-0.01 * t) * np.cos(2 * np.pi * 0.0402 * t) + 0.02 * np.random.default_rng(2).standard_normal(1024)
m = koopman_lift(y[:, None], 16).modes()    # modes of the signal's 16-step delay coordinates
# the strongest mode: frequency 0.04022, decay 0.0102 (true 0.0402, 0.010)

Learn a basis once, apply it to later data

rng = np.random.default_rng(1)
mix = np.linalg.qr(rng.standard_normal((40, 3)))[0].T     # 3 hidden sources over 40 channels
gain = np.exp(rng.uniform(-0.5, 0.5, 40))                   # unequal channel noise

def frames(T):
    return (rng.standard_normal((T, 3)) * [6, 4, 3]) @ mix + gain * rng.standard_normal((T, 40))

b = Aperture(frames(600)).basis()   # b.K, b.F == 3, 40
later = frames(600)

A = b.encode(later)                  # (600, 3): three numbers per row instead of forty
resolved, residual = b.split(later)  # resolved + residual == later
b.drift(later).K                     # 0: the basis still spans what arrives
new = np.linalg.qr(rng.standard_normal((40, 1)))[0][:, 0]
b.drift(frames(600) + np.outer(1.5 * rng.standard_normal(600), new)).K   # 1: a new component

Streaming

ap = Aperture(window=512)
for frame in stream:                 # each frame a vector of F channels (numpy or torch)
    ap.update(frame)

ap.rates()                           # the slowest and fastest decay in the stream
s = ap.state()                       # save, and later resume exactly with Aperture.from_state(s)

Every output shown above is what readme_examples.py prints (output).

Statement of need

Choosing how many components a record holds is the first step of a large class of analyses. It is usually done with a rank or threshold the analyst picks, or with a criterion that needs a known noise level, so a pipeline carries a constant tuned on one instrument and silently wrong on another. Entroptics is for records whose instrument you do not control, or many substrates at once: radio-astronomy waterfalls, spectrograms, sensor panels, embedding stacks. It is measured against the standard selectors (the Gavish–Donoho optimal threshold, Wax–Kailath AIC and MDL) in the validation. For dynamic mode decomposition as such, PyDMD and PyKoopman are more complete.

Know its limits

• Noise correlated across channels or along time (a common-mode drift, 1/f noise) is structure as far as the count is concerned. Remove it, or read a noise reference, first.
• A component has to span channels. A component confined to about (1 + √(F/T))² channels or fewer is not counted, however strong: two channels when records are long, nine for 64 samples across 256 channels.
• Very heavy right tails (lognormal, Pareto noise) exceed the false-alarm rate you set, most on long records.
• A weak component beside a dominant one is counted against the whole record's energy, and can fall under the line.

How it works

Entroptics treats the array as a finite optical aperture. The signal's own entropy sets the resolution it is read at; the count of components is taken against the record's own noise — each channel shuffled in time, the record read again, and the count held to the rate at which pure noise would reach it, at your false-alarm rate, with the Tracy–Widom edge as its closed form; and the dynamics are read as an operator whose modes carry each component's frequency, decay and power. Every read is a classical result (Wiener–Khinchin, Abbe/Rayleigh, Tracy–Widom, Koopman) specialised to finite data, and the governing lemmas are machine-checked in Lean 4.

Reproducing the numbers

The tests, the validation suite and every benchmark generate their inputs from a fixed seed, with no download: pytest, python research/validation/run_all.py, and the scripts in research/benchmarks/. The two FRB figure routines read the public CHIME/FRB Catalog 1 waterfalls (how to fetch them).

Documentation

• The guide: every read and what it means, the conventions for inputs and missing data, and what the library guarantees.
• Benchmarks: counting against the standard rank selectors, cost and accuracy against the FFT, and false-alarm rates, each from a committed script.
• The paper — Markdown · HTML · PDF: the derivations, and where every constant comes from.
• API reference and the CHANGELOG.
• Validation: what each read recovers against a planted truth.

Getting help

• Questions and bug reports: open an issue at github.com/Agience/entroptics/issues, with the array shape, the backend and the read you called.
• Security issues: email connect@agience.ai.
• Contributions: see CONTRIBUTING.md; participation is governed by the Code of Conduct.

Star history

Licensed under Apache-2.0 — see LICENSE.md, NOTICE, PATENTS.md and PLEDGE.md.

Declaration of generative AI use

The author used Anthropic's Claude Opus (versions 4.8 and 5) in the preparation of this work. Its contribution was to write code, and to generate and validate content. The ideas, the construction and the claims are the author's. No other generative AI tool was used. The author reviewed and edited all output and takes full responsibility for the content of this publication.