$ R_{ts} \sim E_{tf}W_{fs} $ where $E_{tf} = R_{ts'}C_{s'f}$ $$ C_{s'f} \geq 0, \Sigma_{s'}C_{s'f}=1 $$ $$ W_{fs} \geq 0, \Sigma_{f} W_{fs} = 1 $$
Canonical Sector  Business Lines  Prototypical Examples 
ccyclical  general and specialty retail discretionary goods 
Gap, Macy's, Target 
cenergy  oil and gas services, equipment, operations 
Halliburton, Schlumberger 
cfinancial  banks insurance (except health) 
US Bancorp., Bank of America 
cindustrial  capital goods, basic materials, transport 
Kennametal, RegalBeloit 
cnoncyclical  consumer staples, healthcare  Pepsi, Procter & Gamble 
creal estate  realty investments and operations  Post Properties, Duke Realty 
ctechnology  semiconductors, computers, communication devices 
Cisco, Texas Instruments 
cutility  electric and gas suppliers  Duke Energy, Wisconsin Energy 
8 (3) Factor AA 
Fama and French 


Normalized Data 
11.1% (5.61%)  4.75% 
S&P 500  93.5%  99.4% 
Equal Weighted S&P 
99.0% (97%) 
95.8% 
Neural network with N1 neurons performs almost as well as network with N neurons (suggests Nth dimension thin)
Is it possible to evaporate neurons in the same way as in the systems biology network?
Can the manifold provide insight for generating new ML architectures?
$$ H\approx\sum_k\bigg{(}\frac{\partial y_k^\theta}{\partial\theta_i}\frac{\partial y_k^\theta}{\partial\theta_j}\bigg{)}=J^TJ$$
$$ p(\theta)=\sqrt{g_{\alpha\beta}}=\sqrt{J^TJ}=\prod_i \Sigma_i $$
N1 and Ndimensional descriptions have a boundary relationship: $M_{Nā1} \approx \partial M_{N}$
Transtrum et. al. has explored the relationship between manifold boundaries and emergent model classes
For a hyperball 95.8 % of the the volume is within 10% of the surface.
Conversely, for a 10D sphere with radius 1 and cap height 0.1, the volume of the cap corresponds to a tiny 0.0014%
Jeffrey's Prior $\rightarrow$ Generative Model (e.g. VAE)
InPCA: Katherine Quinn et. al.
$A(sw) = s^{1}\mathcal{A}(s/w^{1/\sigma})$
$\frac{dM}{dh}(hw) = w^{\beta\beta\delta} \frac{d\mathcal{M}}{dh}((hh_c)/w^{\beta\delta})$
$A(sw) = s^{1}\mathcal{A}(s/\Sigma(w))$
$\frac{dM}{dh}(hw) = \eta(w)^{1}\frac{d\mathcal{M}}{dh}((hh_{max})/\eta(w))$
Power Law Scaling  Lower Critical Dimension  

$\frac{dw}{d\ell}$  $\nu w\ +\ ...$  $w^2+Bw^3$ 
$\xi(w)$  $(\frac{1}{w})^\nu$  $(\frac{1}{w}+B)^{B}exp(\frac{1}{w})$ 
$\frac{ds}{dl}$  $\frac{1}{\sigma\nu}s$  $d_f s  C s w$ 
$\frac{dh}{dl}$  $\frac{\beta\delta}{\nu} h$  $\lambda_h h + F h w$ 
Thank You 

Ricky Chacra Lyft 
Alex Alemi Google AI 
Paul Ginsparg Cornell 
James Sethna Cornell 
Archishman Raju Rockefeller 