Research

Symmetry Analysis & Equivariant Learning

Many visual patterns aren't arbitrary — they're structured by symmetry: an object looks the same after a reflection, a rotation, or some other transformation. Building models that can detect this structure, or that inherently respect it, is a distinct research problem from ordinary recognition, because standard convolutional and attention architectures have no built-in notion of "this feature should transform predictably when the input does." The lab has a sustained line of work addressing this gap from two complementary angles: finding where symmetry exists in an image or shape, and building network architectures that are structurally equivariant to transformations like rotation and reflection by design.

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The symmetry detection strand asks where the axes or planes of reflection and rotation symmetry lie within a single image or 3D shape — treating it as a kind of self-correspondence problem, since detecting symmetry means matching a pattern against a transformed copy of itself. This work progresses from a polar matching formulation for 2D reflection symmetry, to an equivariant-learning approach that handles both reflection and rotation symmetry within one framework, to leveraging 3D geometric priors to make 2D rotation symmetry detection more reliable, to detecting symmetry axes directly through group-equivariant representations, and extending toward the related problem of few-shot pattern detection more broadly.

The equivariant representation learning strand takes the complementary approach of building the transformation structure directly into the network rather than discovering it after the fact. This includes learning keypoint detectors that are equivariant to orientation through self-supervision, extending rotation-equivariant features into general visual correspondence, learning 3D object orientation in a way that stays stable and consistent under different views through invariant residual learning, making semantic correspondence invariant to 3D pose via local shape transforms, and — most recently — predicting 3D rotations directly as Wigner-D harmonic coefficients in the frequency domain, so that the output representation itself is naturally compatible with SO(3)-equivariant spherical networks rather than being awkwardly bolted onto them.

Across both strands, the shared commitment is treating symmetry and equivariance not as incidental properties to be learned implicitly from data, but as structure that should be explicitly detected or explicitly built into a network's representations and outputs.

Research

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