Complex-valued deep learning

AI for signals where phase carries the information

Radio, radar, MRI and coherent optics all measure complex numbers, an amplitude and a phase. Most neural networks split them into two unrelated real numbers. We build networks that compute in complex numbers end to end, and read inputs of any length with a fixed, small memory.

See the research Read the blog

Background: two point sources interfering. Colour is the phase of the field, brightness its strength, the way complex functions are drawn in physics.

What we build

Three ideas, one stack

Complex end to end

Every weight and activation is a complex number, with exact Wirtinger gradients. Phase is kept, not reconstructed.

Born attention

Attention scores come from the rule quantum mechanics uses for measurement probabilities, the squared magnitude of a complex inner product.

Constant-memory streaming

Born attention runs as a fixed-size recurrence, so a model reads millions of samples in one pass without its memory growing.

Where phase matters

Signals that are complex by nature

Many sensors record amplitude and phase, and many of them never stop. A model that keeps phase and listens indefinitely in fixed memory fits both properties.

Radio spectrum monitoringIQ samples: modulation, frequency offsets, emitter fingerprints

Radar and SARComplex returns: range, Doppler velocity, surface structure

Medical imaging (MRI)k-space: spatial position, flow, tissue properties

Coherent optics and fibreOptical field: distance, strain, vibration

Ultrasound and sonarBaseband echoes: depth, motion, direction

Radio interferometryVisibilities: source position and structure

Measured so far

Early evidence

All numbers come from our own runs on public data. They show a working stack and a fast-improving architecture, not general superiority over real-valued models.

12.8 MB

Fixed state of a 12M-parameter complex model reading a 41 MB book test set in 5-million-byte windows, in one pass each.

Useful memory is currently a few thousand bytes; extending it is planned.

0.079 → 0.056

Gap in bits per byte to a standard Transformer of the same size on the same text, closed by one fix in a few days.

Text has no phase; tuning has not started yet.

0.597

Radio modulation recognition on the public RML22 dataset, 11 classes, from raw IQ samples.

Not yet a matched comparison with real-valued models.

Open source

Built on CNet

Our complex-valued framework is open source under the MIT licence: complex layers, Born attention, GPU kernels checked against CPU references, and a one-file API to train your own model.

github.com/crasmarum/CNet →