A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable.
arXiv:2606. 29176v1 Announce Type: new Abstract: A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation.
arXiv:2608. 05136v1 Announce Type: new Abstract: Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not.
arXiv:2606. 03003v1 Announce Type: cross Abstract: A latent world model built from an equivariant encoder $E$ and an equivariant predictor $f$ inherits a provable symmetry of its training loss: when the world's dynamics genuinely carries a group $G$ acting on latents by an orthogonal representation $\rho(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting the dynamics on a restricted slice of orientations mathematically determines it on the entire orbit (j\v{u} y\=i f\v{a}n s\=an).
arXiv:2606. 19491v1 Announce Type: new Abstract: Pretrained transformers sit near singular minima of the loss, where the Fisher information metric degenerates along dead directions: directions in parameter space along which the directional Fisher vanishes.
arXiv:2607. 00603v1 Announce Type: new Abstract: We give a descent-free, alignment-free measurement of singular structure on trained networks.
arXiv:2606. 05957v1 Announce Type: new Abstract: Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate.
arXiv:2607. 12501v3 Announce Type: replace Abstract: The Forward-Forward algorithm trains each layer locally, so that a scalar goodness - the sum of squared activations - is high on real inputs and low on contrastive ones.
arXiv:2606. 31963v1 Announce Type: new Abstract: Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments.
arXiv:2608. 10251v1 Announce Type: cross Abstract: A transformer's answer lives on one axis: the direction its unembedding reads.
arXiv:2605. 18106v3 Announce Type: replace-cross Abstract: A striking geometric disparity has long persisted in the practice of deep learning.
arXiv:2607. 08946v1 Announce Type: new Abstract: A transformer can be built from operators that are legible by construction -- bounded, named units that read as fuzzy set operations rather than dense activations -- but legibility must be pressed for during training, and the pressure has a failure mode.
arXiv:2606. 25971v1 Announce Type: new Abstract: Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object.