Leona(@HiokiLeona):Update on Complex Neural Networks: The Phase-Coherent Transformer (PCT) has solved PathX. PathX is regarded as an extremely difficult long-range context problem even among machine learning tasks; fewer than one in a thousand models/LLMs—including practical ones—can solve it. Even state-of-the-art frontier models cannot solve it directly. This time, a variant of PCT—complexified Screening (titled “Screening is Enough”) augmented with phase-preserving complex diagonal matrix operations—solved PathX with 92% accuracy, once again confirming that preserving phase information forms an effective inductive bias. TBH, the fact that models which do not preserve phase information dropped out of PathX as complex NNs may be more important than the fact that PathX was solved itself. PCT (complex Screening + PCR) is the first purely complex neural network to solve PathX. The original model from which the diagonal recurrence was borrowed was only about 60% complex-valued. Confidence in the claim that “a Transformer for complex NNs, which had not existed until now, has been found” has been strengthening through various experiments. Looking ahead, I expect this family of models to become an entirely new class of models that can simultaneously achieve deep understanding of natural language text and time-series/signal analysis. paper： https://t.co/AMkQzEejYc (note: the latest version is in July) model:

2026.07.29 15:05

Update on Complex Neural Networks: The Phase-Coherent Transformer (PCT) has solved PathX. PathX is regarded as an extremely difficult long-range context problem even among machine learning tasks; fewer than one in a thousand models/LLMs—including practical ones—can solve it. Even state-of-the-art frontier models cannot solve it directly. This time, a variant of PCT—complexified Screening (titled “Screening is Enough”) augmented with phase-preserving complex diagonal matrix operations—solved PathX with 92% accuracy, once again confirming that preserving phase information forms an effective inductive bias. TBH, the fact that models which do not preserve phase information dropped out of PathX as complex NNs may be more important than the fact that PathX was solved itself. PCT (complex Screening + PCR) is the first purely complex neural network to solve PathX. The original model from which the diagonal recurrence was borrowed was only about 60% complex-valued. Confidence in the claim that “a Transformer for complex NNs, which had not existed until now, has been found” has been strengthening through various experiments. Looking ahead, I expect this family of models to become an entirely new class of models that can simultaneously achieve deep understanding of natural language text and time-series/signal analysis. paper： (note: the latest version is in July) model:

Leona@HiokiLeona

2026.05.13 16:09

I was really lucky to come across a very interesting phenomenon in complex neural networks, so I wanted to share it with you. It’s called the Phase Coherent Transformer. Under specific conditions, complex-valued neural networks (CvNNs) exhibit significantly better generalization and superior performance across multiple tasks. I’ve written a detailed experimental report in the paper.Until now, CvNNs were thought to be good mainly in a narrow range of tasks like signal analysis and phase information processing. After Transformers and attention mechanisms came along, the gap with real-valued networks only kept widening. In fact, in this experiment as well, turning a standard Transformer into a complex version mostly just led to performance degradation (except for phase-related tasks). But now, This transformer seems really suited for complex neural networks, unlike before, and it’s showing completely unexpected behaviors and impressive benchmark results. I suspected that “the softmax in Transformers might be causing the loss of phase information in complex weights across multiple layers.” So I looked for attention structures that can preserve phase information, and ran experiments. Approaches like Sigmoid attention and the recently announced Multiscreen have this kind of structure. After complexifying them and tuning for better multi-layer phase control, I named the resulting model the Phase Coherent Transformer (PCT).The experimental results showed that it greatly outperforms many real-valued NNs on tasks where complex NNs had traditionally struggled, while still retaining its advantage on phase-related tasks. Paper preprint: It’s not yet clear how this will impact AI overall, but I think it’s a pretty significant development for the field of complex neural networks at least.This naturally brings up one important question: Q: Should today’s AI stay based on real numbers? In my field, circuit-style cryptography (ZKP, WE), the algebraic structures used as function inputs have naturally expanded into extension fields when necessary. It wouldn’t be surprising if AI follows a similar natural path. Moreover, CvNNs can likely associate and infer from data (=responses and signals from circuits or functions) under the assumption that phase information is hidden inside them. This feels incredibly powerful, because it could allow much more sophisticated handling not only of circuit outputs but also of intermediate state traces.

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