McGan: Mean and Covariance Feature Matching GAN

Mroueh Y, Sercu T, Goel V, et al. McGan: Mean and Covariance Feature Matching GAN[J]. arXiv: Learning, 2017.

@article{mroueh2017mcgan:,
title={McGan: Mean and Covariance Feature Matching GAN},
author={Mroueh, Youssef and Sercu, Tom and Goel, Vaibhava},
journal={arXiv: Learning},
year={2017}}

概

利用均值和協方差構建IPM, 獲得相應的mean GAN 和 covariance gan.

主要內容

IPM:
$d_{\mathscr{F}} (\mathbb{P}, \mathbb{Q}) = \sup_{f \in \mathscr{F}} |\mathbb{E}_{x \sim \mathbb{P}} f(x) - \mathbb{E}_{x \sim \mathbb{Q}} f(x)|.$
當$\mathscr{F}$是對稱空間, 即$f\in \mathscr{F} \rightarrow -f \in \mathscr{F}$,可得
$d_{\mathscr{F}} (\mathbb{P}, \mathbb{Q}) = \sup_{f \in \mathscr{F}} \big \{\mathbb{E}_{x \sim \mathbb{P}} f(x) - \mathbb{E}_{x \sim \mathbb{Q}} f(x) \big\}.$

Mean Matching IPM

$\mathscr{F}_{v,w,p}:= \{f(x)=\langle v, \Phi_w(x) \rangle | v\in \mathbb{R}^m, \|v\|_p \le 1, \Phi_w:\mathcal{X} \rightarrow \mathbb{R}^m, w \in \Omega\},$
其中$\|\cdot \|_p$表示$\ell_p$範數, $\Phi_w$往往用網絡來表示, 我們可通過截斷$w$來使得$\mathscr{F}_{v,w,p}$爲有界線性函數空間(有界從而使得後面推導中$\sup$成爲$\max$).

其中
$\mu_w(\mathbb{P})= \mathbb{E}_{x \sim \mathbb{P}} [\Phi_w(x)] \in \mathbb{R}^m.$

最後一個等式的成立是因爲:
$\|x\|_* = \max \{\langle v, x \rangle | \|v\| \le 1\},$
又$\| \cdot \|_p$的對偶範數是$\|\cdot\|_q, \frac{1}{p}+\frac{1}{q}=1$.

prime

整個GAN的訓練過程即爲
$\tag{3} \min_{g_\theta} \max_{w \in \Omega} \max_{v, \|v\|_p \le 1} \mathscr{L}_{\mu} (v,w,\theta),$
其中
$\mathscr{L}_{\mu} (v,w,\theta) = \langle v, \mathbb{E}_{x \in \mathbb{P}_r} \Phi_w(x) - \mathbb{E}_{z \sim p(z)} \Phi_w(g_{\theta} (z)) \rangle.$

估計形式爲

dual

也有對應的dual形態
$\tag{4} \min_{g_\theta} \max_{w \in \Omega} \|\mu_w(\mathbb{P}_r) - \mu_w (\mathbb{P}_{\theta})\|_q.$

Covariance Feature Matching IPM

$\mathscr{F}_{U, V,w} := \{f(x)= \sum_{j=1}^k \langle u_j, \Phi_w(x) \rangle \langle v_j, \Phi_w(x)\rangle, \langle u_i, u_j \rangle = \langle v_i, v_j \rangle =0, i \not = j, else \:1 \},$
等價於
$\mathscr{F}_{U, V,w} := \{f(x)= \langle U^T \Phi_w(x), V^T\Phi_w(x) \rangle, U^TU=I_k, V^TV=I_k, w \in \Omega \}.$

並有

其中$[A]_k$表示$A$的$k$階近似, 如果$A = \sum_i \sigma_iu_iv_i^T$, $\sigma_1\ge \sigma_2,\ldots$, 則$[A]_k=\sum_{i=1}^k \sigma_i u_iv_i^T$. $\mathcal{O}_{m,k} := \{M \in \mathbb{R}^{m \times k} | M^TM = I_k \}$, $\|A\|_*=\sum_i \sigma_i$表示算子範數.

prime

$\tag{6} \min_{g_\theta} \max_{w \in \Omega} \max_{U,V \in \mathcal{P}_{m, k}} \mathscr{L}_{\sigma} (U, V,w,\theta),$
其中
$\mathscr{L}_{\sigma} (U,V,w,\theta) = \mathbb{E}_{x \sim \mathbb{P}_r} \langle U^T \Phi_w(x), V^T\Phi_w(x) \rangle- \mathbb{E}_{z \sim p_z} \langle U^T \Phi_w(g_{\theta}(z)), V^T\Phi_w(g_{\theta}(z)) \rangle.$

採用下式估計

dual

$\tag{7} \min_{g_{\theta}} \max_{w \in \Omega} \| [\Sigma_w(\mathbb{P}_r) - \Sigma_w(\mathbb{P}_{\theta})]_k\|_*.$

注: 既然$\Sigma_w(\mathbb{P}_r) - \Sigma_w(\mathbb{P}_{\theta})$是對稱的, 爲什麼$U \not =V$? 因爲雖然其對稱, 但是並不(半)正定, 所以$v_i=-u_i$也是有可能的.

算法

代碼

未經測試.

import torch
import torch.nn as nn
from torch.nn.functional import relu
from collections.abc import Callable



def preset(**kwargs):
    def decorator(func):
        def wrapper(*args, **nkwargs):
            nkwargs.update(kwargs)
            return func(*args, **nkwargs)
        wrapper.__doc__ = func.__doc__
        wrapper.__name__ = func.__name__
        return wrapper
    return decorator


class Meanmatch(nn.Module):

    def __init__(self, p, dim, dual=False, prj='l2'):
        super(Meanmatch, self).__init__()
        self.norm = p
        self.dual = dual
        if dual:
            self.dualnorm = self.norm
        else:
            self.init_weights(dim)
            self.projection = self.proj(prj)


    @property
    def dualnorm(self):
        return self.__dualnorm

    @dualnorm.setter
    def dualnorm(self, norm):
        if norm == 'inf':
            norm = float('inf')
        elif not isinstance(norm, float):
            raise ValueError("Invalid norm")

        p = 1 / (1 - 1 / norm)
        self.__dualnorm = preset(p=p, dim=1)(torch.norm)


    def init_weights(self, dim):
        self.weights = nn.Parameter(torch.rand((1, dim)),
                                    requires_grad=True)

    @staticmethod
    def _proj1(x):
        u = x.max()
        if u <= 1.:
            return x
        l = 0.
        c = (u + l) / 2
        while (u - l) > 1e-4:
            r = relu(x - c).sum()
            if r > 1.:
                l = c
            else:
                u = c
            c = (u + l) / 2
        return relu(x - c)

    @staticmethod
    def _proj2(x):
        return x / torch.norm(x)

    @staticmethod
    def _proj3(x):
        return x / torch.max(x)

    def proj(self, prj):
        if prj == "l1":
            return self._proj1
        elif prj == "l2":
            return self._proj2
        elif prj == "linf":
            return self._proj3
        else:
            assert isinstance(prj, Callable), "Invalid prj"
            return prj



    def forward(self, real, fake):
        temp = (real - fake).mean(dim=1)
        if self.dual:
            return self.dualnorm(temp)
        elif not self.training and self.dual:
            raise TypeError("just for training...")
        else:
            self.weights.data = self.projection(self.weights.data) #some diff here!!!!!!!!!!
            return self.weights @ temp



class Covmatch(nn.Module):

    def __init__(self, dim, k):
        super(Covmatch, self).__init__()
        self.init_weights(dim, k)

    def init_weights(self, dim, k):
        temp1 = torch.rand((dim, k))
        temp2 = torch.rand((dim, k))
        self.U = nn.Parameter(temp1, requires_grad=True)
        self.V = nn.Parameter(temp2, requires_grad=True)

    def qr(self, w):
        q, r = torch.qr(w)
        sign = r.diag().sign()
        return q * sign

    def update_weights(self):
        self.U.data = self.qr(self.U.data)
        self.V.data = self.qr(self.V.data)

    def forward(self, real, fake):
        self.update_weights()
        temp1 = real @ self.U
        temp2 = real @ self.V
        temp3 = fake @ self.U
        temp4 = fake @ self.V
        part1 = torch.trace(temp1 @ temp2.t()).mean()
        part2 = torch.trace(temp3 @ temp4.t()).mean()
        return part1 - part2