SVCCA

(Raghu et al., 2017) proposed a way to compare two representations that is both invariant to affine transform and fast to compute ¹

based on canonical correlation analysis which was invariant to linear transformation.

definition

Given a dataset $X = \{x_{1},\cdots, x_m\}$ and a neuron $i$ on layer $l$, we define $z_i^l$ to be the vector of outputs on $X$, or:

$$z^l_i = (z^l_i(x_1), \cdots, z^l_i(x_m))$$

SVCCA proceeds as following:

1. Input: takes as input two (not necessary different) sets of neurons $l_{1} = \{z_1^{l_{1}}, \cdots, z_{m_{1}}^{l_1}\}$ and $l_{2} = \{z_1^{l_2}, \cdots, z_{m_2}^{l_{2}}\}$

2. Step 1: Perform SVD of each subspace to get subspace $l^{'}_1 \subset l_1, l^{'}_2 \subset l_2$

3. Step 2: Compute Canonical Correlation similarity between $l^{'}_1, l^{'}_2$, that is maximal correlations between $X,Y$ can be expressed as:

   $$\max \frac{a^T \sum_{XY}b}{\sqrt{a^T \sum_{XX}a}\sqrt{b^T \sum_{YY}b}}$$

   where $\sum_{XX}, \sum_{XY}, \sum_{YX}, \sum_{YY}$ are covariance and cross-variance terms.

   By performing change of basis $\tilde{x_{1}} = \sum_{xx}^{\frac{1}{2}} a$ and $\tilde{y_1}=\sum_{YY}^{\frac{1}{2}} b$ and Cauchy-Schwarz we recover an eigenvalue problem:

   $$\tilde{x_{1}} = \argmax [\frac{x^T \sum_{X X}^{\frac{1}{2}} \sum_{XY} \sum_{YY}^{-1} \sum_{YX} \sum_{XX}^{-\frac{1}{2}}x}{\|x\|}]$$

4. Output: aligned directions $(\tilde{z_i^{l_{1}}}, \tilde{z_i^{l_{2}}})$ and correlations $\rho_i$

distributed representations

SVCCA has no preference for representations that are neuron (axed) aligned. ²