Machine Learning.

Introduction

(Notes from a reading of Cohen and Welling: Steerable CNNs.)

Consider \(\mathbb{R}^2\) as an affine hyperplane in \(\mathbb{R}^3\), embedded via the map \(x\mapsto (x,1)\). Then the Euclidean motion group \(\tilde G = \mathbb{R}^2\ltimes O(2)\) has a convenient matrix representation. Let \(r\) be a rotation and \(t\) a translation. Then

\[ \begin{split} r &= \begin{pmatrix} R & 0 \\ 0 & 1 \\ \end{pmatrix} \\\
t &= \begin{pmatrix} I & T \\ 0 & 1 \\ \end{pmatrix} \end{split} \]

where \(R\in O(2)\) and \(T\in \mathbb{R}^2\). Given \(x\in \mathbb{R}^2\), we may identify it with a translation map in \(\tilde G\) via

\[\bar x = \begin{pmatrix} I & x \\ 0 & 1 \\ \end{pmatrix}.\]

In image recognition, \(\mathbb{Z}^2\) parametrizes the pixels of an image with infinite width and height, and a discrete subgroup \(G\) of \(\tilde G\) acts on this parametrization. In particular, \(G = \mathbb{Z}^2\ltimes D_4\), so that our parametrization is the homogeneous space \(\mathbb{Z}^2 = G/D_4\). Let’s call this parametrization the pixel space.

Images are described by feature maps, which are functions \(f:G/D_4 \rightarrow \mathbb{R}^K\), with each dimension of the target interpreted as a color channel. For example, \(K=3\) may correspond to an RGB image, and \(K=4\) may correspond to a CMYK image. A feature map typically takes non-negative rational values, and these values correspond to pixel intensities. A representation of a real-world image would thus be a compactly supported feature map.

Over the pixel space, we have a homogeneous vector bundle \(G\times_{D_4} \mathbb{R}^K\) with an action by the discrete motion group given by \(g'\cdot(g,v) = (g'g, v)\). Let \(\mathcal{F} = \Gamma(G\times_{D_4} \mathbb{R}^K)\) be the space of all feature maps. \(G\) acts on it by left translation: \(\pi(tr) f(xD_4) = (tr)\cdot f((tr)^{-1}\cdot xD_4).\)

Let \(\Psi: \mathcal{F}\rightarrow \mathbb{R}^{K’}\) be a filter bank. This can be thought of as a collection of \(K'\) linear functionals on the space of feature maps. Each linear functional is an operation that is performed on the value of a fixed pixel (which we may as well assume is the origin). Each linear functional outputs a weighted sum of the value associated to the fixed pixel and its neighbors. By translating pixels to the origin, we can construct a feature map \(\Psi\ast f\in \mathcal{F}’\), where \(\mathcal{F}’ = \Gamma(G\times_{D_4} \mathbb{R}^{K’})\) is another space of feature maps. This is defined as follows:

\[(\Psi\ast f)(x) = \Psi(\pi(\bar x)^{-1} f).\]

Thus we get a map \(\Phi:\mathcal{F}\rightarrow \mathcal{F}’\) via \(\Phi(f) = \Psi\ast f\). This map is called a convolutional neural network.

Let \((\mathbb{R}^{K’},\rho)\) be a representation of \(D_4\) on the model fiber of \(\mathcal{F}’\). Our objective is to construct a representation \((\mathcal{F}’,\pi’)\) such that if the filter bank \(\Psi\) intertwines the dihedral group representations \((\mathcal{F},\pi)\) and \((\mathbb{R}^{K'},\rho)\), then the convolutional neural network \(\Phi\) intertwines the discrete motion group representations \((\mathcal{F},\pi)\) and \((\mathcal{F}',\pi')\). First we need some algebra:

Lemma. Let \(r\) be a rotation, \(t\) a translation, and \(\bar x\) the translation corresponding to the position \(x\in\mathbb{R}^2\). Then

\[(tr)^{-1} \bar x^{-1} r = \overline{(tr)^{-1}\cdot x}.\]

The proof is a straightforward application of the given matrix representation. This result leads to the following equivariance rule.

Proposition. Let \(r\) be a rotation, \(t\) a translation, and \(\bar x\) the translation corresponding to the position \(x\in\mathbb{Z}^2\). If \(\Psi\pi(r) = \rho(r)\Psi\) for all \(r\in D_4\), then

\[(\Psi\ast \pi(tr) f)(x) = \rho(r)(\Psi\ast f)((tr)^{-1}\cdot x).\]

Proof. By using an identity trick, we can exploit our two versions of the same map to produce the equivariance law:

\[ \begin{split} (\Psi\ast \pi(tr) f)(x) &= \Psi(\pi(\bar x)^{-1}\pi(tr)f)\\ &= \rho(r)\Psi(\pi(r)\pi(r)^{-1}\pi(\bar x)^{-1}\pi(tr)f) \\ &= \rho(r)\Psi(\pi(r^{-1}\bar x^{-1}tr)f)\\ &= \rho(r)\Psi(\pi((tr)^{-1}\bar xr)^{-1}f)\\ &= \rho(r)\Psi(\pi(\overline{(tr)^{-1}\cdot x})^{-1}f)\\ &= \rho(r)(\Psi\ast f)((tr)^{-1}\cdot x) \end{split} \]

With this calculation in mind, we are now in a position to define a representation of \(G\) on \(\mathcal{F}'\):

\[\pi'(tr)(\Psi\ast f)(x) = \rho(x)(\Psi\ast f)((tr)^{-1}\cdot x).\]

To verify this, we note that one checks the relation \(f(t_1r_1t_2r_2) = f(t_1r_1)f(t_2r_2)\) by using the fact that the conjugate of a translation by a rotation is again a translation, and that

\[t_1r_1t_2r_2 = t_1r_1t_2r_1^{-1}r_1r_2.\]

As a consequence, we get an intertwining property:

Corollary. If \(\Psi\pi(r) = \rho(r)\Psi\) for all \(r\in D_4\), then \(\Phi\, \pi(g) = \pi'(g)\, \Phi\) for all \(g\in G\).

Whenever we can find representations \((\mathcal{F},\pi)\) and \((\mathcal{F}',\pi')\) for which \(\Phi\) is an intertwiner, we say that \(\Phi\) is a steerable convolutional neural network.

To determine the homogeneous vector bundle of which \(\Psi\ast f\) is a section, it is enough to calculate the action of \(D_4\) on \((\Psi\ast f)(0)\):

\[\pi’(r)(\Psi\ast f)(0) = \rho(r)(\Psi\ast f)(\pi(r)^{-1}\cdot 0) = \rho(r)(\Psi\ast f)(0).\]

This means that \(\Psi\ast f\in \Gamma(G\times_\rho \mathbb{R}^{K'})\). From representation theory, we know that

\[\Gamma(G\times_\rho \mathbb{R}^{K'}) \cong Ind_{D_4}^G(\rho)\]

thus we may interpret \(\Phi\) as a map into an induced representation of the discrete motion group. Moreover, we can treat \(\Gamma(G\times_{D_4} \mathbb{R}^K)\) as an induced representation as well.

On \(\mathcal{F}\), the action of \(D_4\) should only rotate the pixel of the image; there should be no linear transformations within fibers (i.e. no transformations of color channels). This means that \(D_4\) has a trivial action on the value of \(f\) at the origin. Hence we are regarding \(\mathcal{F}\) as \(\Gamma(G\times_{\rho_0} \mathbb{R}^K)\), where \(\rho_0\) denotes the trivial representation of \(D_4\). To summarize, \[\Gamma(G\times_{\rho_0} \mathbb{R}^K)\cong Ind_{D_4}^G(\rho_0)\cong C(\mathbb{Z}^2).\] We can now restate this picture in a representation theoretic context.

Theorem. Let \((\mathbb{R}^K,\rho_0)\) be a trivial representation of \(D_4\) and \(Ind_{D_4}^G(\rho_0)\) a space of feature maps. Let \(\Psi : Ind_{D_4}^G(\rho_0) \rightarrow \mathbb{R}^{K'}\) be a \(D_4\)-equivariant filter with respect to \((\mathbb{R}^{K'},\rho)\). Then \[\Phi: Ind_{D_4}^G(\rho_0) \rightarrow Ind_{D_4}^G(\rho)\] is a steerable convolutional neural network.