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Part2.5-ASharpGeneralizationBound
Orbit directions are trivially flat, inflating sharpness estimates. Quotient-space sharpness factors out reparametrization symmetry for tighter generalization bounds.
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Part3.5-PERecoveryProof
A complete proof that classical group equivariance is recovered from path equivariance under the endpoint condition, establishing classical equivariant networks as a special case of the PEN framework.
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PML-1 MAP MLE KL
MAP vs MLE as point estimates, global parameters vs per-example latent variables, the ELBO, and why the two directions of KL divergence give fundamentally different approximations.
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Part5-PENHolonomyandSingleTangentFallacy
Path equivariant networks via parallel transport, holonomy-controlled expressivity, and why the single tangent space approach fails on curved manifolds.
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TheWorldFromWithinAndWithout
Intrinsic and extrinsic perspectives in mathematics, physics, and philosophy.
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Semidefinite Programming and Applications
SDP formulation, duality, and applications to Euclidean distance completion and sparse PCA.
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Independence in Bayesian Network Causal Diagrams
Independence and conditional independence in Bayesian networks, d-separation, and the collider effect.
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PML-2 From Likelihood to ELBO
The probabilistic ML pipeline: notation, likelihood, ELBO derivation, and the reparameterization trick for VAEs.
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From Distances to Coordinates (Euclidean)
Recovering point coordinates from pairwise distances via Gram matrices and eigendecomposition.
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Yoneda Perspective
Understanding the Yoneda Lemma and its deep implications in category theory.
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Part1-PriorKnowledge
Prior knowledge in neural networks: every design choice encodes a structural assumption about the world.
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Part3-PathEquivariance
Generalizing group equivariance to path equivariance on manifolds, with fiber bundles and the content-pose decomposition.
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Part2-GroupStructure
How activation functions and regularization break the symmetry group of deep networks, traced from GL(n) through specific subgroups.
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Part4-CategoryTheoryPerspective
Equivariance is naturality: unifying groups, manifolds, and path equivariance through category theory.
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Pushforward Pullback
Pushforward and pullback in differential geometry and probability, with the duality between vectors and forms.
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What is dx?
What dx means: from calculus infinitesimals to differential 1-forms on manifolds.
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Category Product
The categorical product via universal property, with arguments about arrow direction and coproducts.
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Product
Products across mathematics: inner/outer/cross products, Kronecker, group products, tensor products, and their universal properties.
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Probability-0
Probability spaces, conditional probability, random variables, independence, and expectation from measure theory.
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PCA
PCA as eigendecomposition of the covariance matrix, its SVD implementation, and why it works.