Objects are represented in each stage of the sensory hierarchy as manifolds due to variabilities in
stimulus features such as location, orientation, and intensity. What makes manifold representations at
higher stages of the hierarchy better suited for invariant decoding of object information by downstream
circuits than earlier stages? It has been suggested that the sensory hierarchy becomes increasingly untangled,
but the notion of “tangled manifolds” remains vague. In this work, we consider linear readout of
objects as a model of biologically plausible computation of sensory signals to determine which statistical
features of the neuronal representation of object manifolds are more amenable to this computation.
Extending the statistical mechanics of linear classification of random points, we establish a theory of
linear classification of generic manifolds synthesizing statistical and geometric properties of high dimensional
signals. The exact capacity of manifold classification generally depends on the full geometrical
details of their convex hulls; however, we show that in a broad parameter regime, manifold classification
depends primarily on two quantities: their e↵ective dimension, Deff , and their e↵ective radius, Reff .
We have developed a novel efficient algorithm that can learn the synaptic weights of the neuronal readout
guaranteed to converge to the solution with good generalization and robustness properties. We demonstrate
results from applying our method to both neuronal networks and deep networks from machine
learning.