f (ρ (g) x) = ρ^{'} (g) f (x)

\overset{x}{^} = U^{⊤} x

L = U Λ U^{⊤}

[X, Y] = X Y - Y X

H^{(l + 1)} = σ (\tilde{D}^{- \frac{1}{2}} \tilde{A} \tilde{D}^{- \frac{1}{2}} H^{(l)} W^{(l)})

L = D - A

exp_{p} : T_{p} M \to M

Attn (Q, K, V) = softmax (\frac{Q K ^{⊤}}{d _{k}}) V

\nabla_{X} Y = \partial_{X} Y + Γ (X, Y)

d s^{2} = g_{ij} (x) d x^{i} d x^{j}

\frac{\partial u}{\partial t} = Δ_{M} u

α_{ij} = \frac{e x p ( e _{ij} )}{\sum _{k \in N_{i}} e x p ( e _{ik} )}

\overset{x}{¨}^{k} + Γ_{ij}^{k} \overset{x}{˙}^{i} \overset{x}{˙}^{j} = 0

k_{t} (x, y) = \sum_{i} e^{- λ_{i} t} φ_{i} (x) φ_{i} (y)

Φ (R x + t) = R Φ (x)

h_{v}^{(l + 1)} = ϕ (h_{v}^{(l)}, ⨁_{u \in N (v)} ψ (h_{u}^{(l)}, h_{v}^{(l)}, e_{uv}))

g_{θ} (L) = \sum_{k = 0}^{K} θ_{k} T_{k} (\tilde{L})

(f ⋆ ψ) (g) = \int_{G} f (h) ψ (g^{- 1} h) d μ (h)

c_{v}^{(l + 1)} = HASH (c_{v}^{(l)}, {{c_{u}^{(l)}}}_{u \in N (v)})

d x_{t} = \nabla lo g p (x_{t}) d t + 2 d W_{t}

p_{X} (x) = p_{Z} (f^{- 1} (x)) det \frac{\partial f ^{- 1}}{\partial x}

L = E_{q} [lo g p (x ∣ z)] - KL (q ∥ p)

Fractal Manifold

Mathematical Deep Learning Research

Research Founders

AI Research

Deep learning has achieved remarkable results, but current architectures are hitting fundamental limits. We believe the path forward lies in mathematics—specifically in understanding the geometric structure of data and the symmetries that govern learning.

Fractal Manifold is a research lab focused on advancing deep learning through rigorous mathematical foundations. We work at the intersection of differential geometry, group theory, and neural network design to build models that are not just bigger, but fundamentally better.

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