# notes

why relevant: powerful class of transforms with nice properties (continuous) for learning represented as differential manifolds, which also interested in Also working on biologically-inspired vision systems mentored by Bruno Olshausen. Lie groups + sparse codes have great emperical results
tooling from topology and abstract algebra
paper notes and derivations
Introduction to smooth manifold optimization&hellip; Notes from this textbook
Formulations that motivate manifold optimization. Logistic regression Taking a well known formulation from linear algebra and teasing out a curvy structure. Consider $x_1,\ldots,x_m \in \xi$ each with a corresponding binary label $y_1,\ldots,y_m \in \{0,1\}$. We can define some vector $\theta \in \xi$ that we can &ldquo;grab onto&rdquo; with each $x_i$ using an inner-product defined over the space. Applying a logistic transform $\sigma$, we can obtained well-behaved probabilities: $$P[y=1|x,\theta] = \sigma(\lang\theta, x_i\rang)$$ $$P[y=0|x,\theta] = 1 - \sigma(\lang\theta, x_i\rang)$$