Self-Conditioning Enables Deep Sequential Reasoning in Flow Matching

Self-Conditioning Enables Deep Sequential Reasoning in Flow Matching. By Zeyang Zhang and six others, from the Shanghai Qi Zhi Institute, Tsinghua University, and two other institutions. Published on the fourteenth of October. Its code repository is worked out: about two hundred and fifty source files, with a readme.

Recent work has tried visual reasoning through image generation: a model solves a visual-reasoning problem by directly generating the image of its solution. Iterative samplers such as flow matching seem a natural fit. Each sampling step might resemble one step of reasoning, so more steps ought to mean more reasoning. The authors examine that intuition from the perspective of circuit complexity, and conclude that it is misleading.

There are two results, a negative one and a positive one. The negative result: for vanilla flow matching, additional sampling steps cannot increase the expressive power of a constant-depth backbone model, under the standard complexity-theoretic model of transformers. The positive result: with self-conditioning, a constant-depth backbone can achieve expressive power equivalent to that of space-bounded Turing machines.

The negative result rests on a simulation argument. Provided the model's outputs satisfy certain mild regularity conditions, the whole trajectory of the vanilla flow-matching sampler can be simulated by a constant number of sequential rounds of parallel model evaluations. However many steps the sampler takes, the computation collapses into a fixed number of rounds of sequential depth. Extra steps add work, but not depth, and so they cannot add expressive power.

Self-conditioning changes what passes from one step to the next. The model is conditioned on its own estimate, from the previous step, of the final image. The authors identify this as playing a previously unrecognized role: it removes the bottleneck. With that estimate carried forward from step to step, the expressive power reaches that of space-bounded Turing machines.

Controlled experiments on diverse visual-reasoning tasks match the theory. Flow matching with self-conditioning achieves higher accuracy and greater reasoning depth than vanilla flow matching, and its performance improves as more sampling steps are added.

The limits sit in the hypotheses. The negative result needs the model outputs to satisfy mild regularity conditions. It concerns a constant-depth backbone. And it is stated under the standard complexity-theoretic model of transformers, so its reach is the reach of that model. The positive result is likewise about expressive power: what a constant-depth backbone with self-conditioning can in principle represent.

The authors draw a design principle from this for future visual reasoning work. Carry more information between sampling steps, so that additional steps translate into deeper reasoning. Steps alone do not make a sampler think longer. What each step hands to the next is what decides it.