网站怎么会k,杭州高端企业网站建设,蜜雪冰城网络营销案例分析,可以做卷子的网站机器学习笔记之流形模型——标准流模型基本介绍引言回顾#xff1a;隐变量模型的缺陷标准流(Normalizing Flow\text{Normalizing Flow}Normalizing Flow)思想分布变换的推导过程引言
本节将介绍概率生成模型——标准流模型(Normalizing Flow\text{Normalizing Flow}Normalizi…
机器学习笔记之流形模型——标准流模型基本介绍引言回顾隐变量模型的缺陷标准流(Normalizing Flow\text{Normalizing Flow}Normalizing Flow)思想分布变换的推导过程引言
本节将介绍概率生成模型——标准流模型(Normalizing Flow\text{Normalizing Flow}Normalizing Flow)。
回顾隐变量模型的缺陷
关于隐变量模型(Latent Variable Model,LVM\text{Latent Variable Model,LVM}Latent Variable Model,LVM)如果表示隐变量的随机变量集合Z\mathcal ZZ足够复杂的话很容易出现积分难问题 此时隐变量Z\mathcal ZZ的维度(随机变量个数)极高(M)(\mathcal M)(M),对Z\mathcal ZZ求解积分的代价是极大的(Intractable)(\text{Intractable})(Intractable). P(X)⏟Intractable∫ZP(Z,X)dZ∫ZP(Z)⋅P(X∣Z)dZ∫Z1⋯∫ZMP(Z1,⋯,ZM)⋅P(X∣Z1,⋯,ZM)dZ1,⋯,ZM\begin{aligned} \underbrace{\mathcal P(\mathcal X) }_{\text{Intractable}} \int_{\mathcal Z} \mathcal P(\mathcal Z,\mathcal X) d\mathcal Z \\ \int_{\mathcal Z} \mathcal P(\mathcal Z) \cdot \mathcal P(\mathcal X \mid \mathcal Z) d\mathcal Z \\ \int_{\mathcal Z_1} \cdots \int_{\mathcal Z_{\mathcal M}} \mathcal P(\mathcal Z_1,\cdots,\mathcal Z_{\mathcal M}) \cdot \mathcal P(\mathcal X \mid \mathcal Z_1,\cdots,\mathcal Z_{\mathcal M}) d\mathcal Z_1,\cdots,\mathcal Z_{\mathcal M} \end{aligned}IntractableP(X)∫ZP(Z,X)dZ∫ZP(Z)⋅P(X∣Z)dZ∫Z1⋯∫ZMP(Z1,⋯,ZM)⋅P(X∣Z1,⋯,ZM)dZ1,⋯,ZM 从而关于隐变量Z\mathcal ZZ的后验概率P(Z∣X)\mathcal P(\mathcal Z \mid \mathcal X)P(Z∣X)也同样是极难求解的 P(Z∣X)⏟IntractableP(Z,X)P(X)P(Z)⋅P(X∣Z)P(X)⏟Intractable\begin{aligned} \underbrace{\mathcal P(\mathcal Z \mid \mathcal X)}_{\text{Intractable}} \frac{\mathcal P(\mathcal Z,\mathcal X)}{\mathcal P(\mathcal X)} \\ \frac{\mathcal P(\mathcal Z) \cdot \mathcal P(\mathcal X \mid \mathcal Z)}{\underbrace{\mathcal P(\mathcal X)}_{\text{Intractable}}} \end{aligned}IntractableP(Z∣X)P(X)P(Z,X)IntractableP(X)P(Z)⋅P(X∣Z)
针对这种问题由于无法得到精确解/精确解计算代价极高因而通常采用近似推断(Approximate Inference\text{Approximate Inference}Approximate Inference)的方式对P(Z∣X)\mathcal P(\mathcal Z \mid \mathcal X)P(Z∣X)近似求解。
例如变分自编码器(Variational Auto-Encoder,VAE\text{Variational Auto-Encoder,VAE}Variational Auto-Encoder,VAE)它的底层逻辑是使用重参数化技巧将人为设定分布Q(Z∣X)\mathcal Q(\mathcal Z \mid \mathcal X)Q(Z∣X)视作关于参数ϕ\phiϕ的函数Q(Z∣X,ϕ)\mathcal Q(\mathcal Z \mid \mathcal X,\phi)Q(Z∣X,ϕ)并通过神经网络学习参数ϕ\phiϕ并使其近似P(Z∣X)\mathcal P(\mathcal Z \mid \mathcal X)P(Z∣X)。关于变分自编码器的模型结构表示如下 关于编码器(Encoder\text{Encoder}Encoder)函数Q(Z∣X;ϕ)\mathcal Q(\mathcal Z \mid \mathcal X;\phi)Q(Z∣X;ϕ)与解码器(Decoder\text{Decoder}Decoder)函数P(X∣Z;θ)\mathcal P(\mathcal X \mid \mathcal Z;\theta)P(X∣Z;θ)变分自编码器的目标函数表示如下 一个有趣的现象其中−KL[Q(Z∣X;ϕ)∣∣P(Z;θ(t))]- \text{KL} [\mathcal Q(\mathcal Z \mid \mathcal X;\phi) || \mathcal P(\mathcal Z ;\theta^{(t)})]−KL[Q(Z∣X;ϕ)∣∣P(Z;θ(t))]只是一个关于ϕ\phiϕ的惩罚项(约束)并且这个约束直接作用于EQ(Z∣X;ϕ)[logP(X∣Z;θ)]\mathbb E_{\mathcal Q(\mathcal Z \mid \mathcal X;\phi)} \left[\log \mathcal P(\mathcal X \mid \mathcal Z;\theta)\right]EQ(Z∣X;ϕ)[logP(X∣Z;θ)].因此真正迭代的只有参数θ(θ(t)⇒θ(t1))\theta(\theta^{(t)}\Rightarrow \theta^{(t1)})θ(θ(t)⇒θ(t1)),参数ϕ\phiϕ仅是迭代过程中伴随着θ\thetaθ的更新而更新。 {L(ϕ,θ,θ(t))EQ(Z∣X;ϕ)[logP(X∣Z;θ)]−KL[Q(Z∣X;ϕ)∣∣P(Z;θ(t))](θ^(t1),ϕ^(t1))argmaxθ,ϕL(ϕ,θ,θ(t))\begin{cases} \mathcal L(\phi,\theta,\theta^{(t)}) \mathbb E_{\mathcal Q(\mathcal Z \mid \mathcal X;\phi)} \left[\log \mathcal P(\mathcal X \mid \mathcal Z;\theta)\right] - \text{KL} [\mathcal Q(\mathcal Z \mid \mathcal X;\phi) || \mathcal P(\mathcal Z;\theta^{(t)})] \\ \quad \\ (\hat {\theta}^{(t1)},\hat {\phi}^{(t1)}) \mathop{\arg\max}\limits_{\theta,\phi} \mathcal L(\phi,\theta,\theta^{(t)}) \end{cases}⎩⎨⎧L(ϕ,θ,θ(t))EQ(Z∣X;ϕ)[logP(X∣Z;θ)]−KL[Q(Z∣X;ϕ)∣∣P(Z;θ(t))](θ^(t1),ϕ^(t1))θ,ϕargmaxL(ϕ,θ,θ(t)) 关于目标函数L(ϕ,θ,θ(t))\mathcal L(\phi,\theta,\theta^{(t)})L(ϕ,θ,θ(t))的底层逻辑是最大化ELBO\text{ELBO}ELBO (θ^(t1),ϕ^(t1))argmaxθ,ϕ{EQ(Z∣X;ϕ)[logP(X,Z;θ)Q(Z∣X;ϕ)]}(\hat {\theta}^{(t1)},\hat {\phi}^{(t1)}) \mathop{\arg\max}\limits_{\theta,\phi} \left\{\mathbb E_{\mathcal Q(\mathcal Z \mid \mathcal X;\phi)} \left[\log \frac{\mathcal P(\mathcal X,\mathcal Z;\theta)}{\mathcal Q(\mathcal Z \mid \mathcal X;\phi)}\right]\right\}(θ^(t1),ϕ^(t1))θ,ϕargmax{EQ(Z∣X;ϕ)[logQ(Z∣X;ϕ)P(X,Z;θ)]} 也就是说它仅仅是最大化了极大似然估计logP(X;θ)\log \mathcal P(\mathcal X;\theta)logP(X;θ)的下界。实际上它并没有直接对对数似然函数求解最优化问题。
这不可避免地存在误差毕竟最优化对数似然函数和最优化它的下界 是两个概念。这一切的核心问题均在于P(X)\mathcal P(\mathcal X)P(X)无法得到精确解。
如果存在一种模型它在学习任务过程中P(X)\mathcal P(\mathcal X)P(X)是可求解的(tractable\text{tractable}tractable)自然不会出现上述一系列的近似操作了。
标准流(Normalizing Flow\text{Normalizing Flow}Normalizing Flow)思想
关于样本X\mathcal XX的概率分布P(X)\mathcal P(\mathcal X)P(X)它可能是复杂的。但流模型(Flow-based Model\text{Flow-based Model}Flow-based Model)的思想是分布P(X)\mathcal P(\mathcal X)P(X)的复杂并不是一蹴而就的而是通过若干次的变化而产生出的复杂结果。
关于流模型的概率图结构可表示为如下形式 从模型结构中可以观察到既然分布P(X)\mathcal P(\mathcal X)P(X)比较复杂那么可以构建隐变量Z\mathcal ZZ与X\mathcal XX之间的函数关系Xf(Z)\mathcal X f(\mathcal Z)Xf(Z)从而通过换元的方式描述P(Z)\mathcal P(\mathcal Z)P(Z)与P(X)\mathcal P(\mathcal X)P(X)的函数关系。
如果隐变量Z\mathcal ZZ的结构同样复杂可以继续针对该隐变量创造新的隐变量并构建函数关系。以此类推最终可以通过一组服从简单分布的随机变量Zinit\mathcal Z_{init}Zinit通过若干次的函数的嵌套表示得到关于X\mathcal XX的关联关系从而得到Pinit(Zinit)⇒P(X)\mathcal P_{init}(\mathcal Z_{init}) \Rightarrow \mathcal P(\mathcal X)Pinit(Zinit)⇒P(X)的函数关系。
分布变换的推导过程
以上图中隐变量ZK\mathcal Z_{\mathcal K}ZK和观测变量X\mathcal XX之间关联关系示例
创建假设fKf_{\mathcal K}fK是一个 连续、可逆 函数满足XfK(ZK)\mathcal X f_{\mathcal K}(\mathcal Z_{\mathcal K})XfK(ZK)。其中ZK,X\mathcal Z_{\mathcal K},\mathcal XZK,X均表示随机变量集合并服从对应的概率分布 其中PX(X)\mathcal P_{\mathcal X}(\mathcal X)PX(X)表示关于X\mathcal XX的概率分布并且变量是X.ZK\mathcal X.\mathcal Z_{\mathcal K}X.ZK对应分布同理。反过来由于fKf_{\mathcal K}fK函数可逆因而有ZKfK−1(X)\mathcal Z_{\mathcal K} f_{\mathcal K}^{-1}(\mathcal X)ZKfK−1(X). ZK∼PZK(ZK),X∼PX(X);ZK,X∈Rp\mathcal Z_{\mathcal K} \sim \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K}),\mathcal X \sim \mathcal P_{\mathcal X}(\mathcal X);\quad \mathcal Z_{\mathcal K},\mathcal X \in \mathbb R^pZK∼PZK(ZK),X∼PX(X);ZK,X∈Rp 不可否认的是无论是PZK(ZK)\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PZK(ZK)还是PX(X)\mathcal P_{\mathcal X}(\mathcal X)PX(X)它们都是概率分布。根据概率密度积分的定义必然有 ∫ZKPZK(ZK)dZK∫XPX(X)dX1\int_{\mathcal Z_{\mathcal K}} \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K}) d\mathcal Z_{\mathcal K} \int_{\mathcal X} \mathcal P_{\mathcal X}(\mathcal X) d\mathcal X 1∫ZKPZK(ZK)dZK∫XPX(X)dX1 从而有 在变分推断——重参数化技巧一节中也使用这种描述进行换元,在不定积分中,PZK(ZK)dZK\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K}) d\mathcal Z_{\mathcal K}PZK(ZK)dZK和PX(X)dX\mathcal P_{\mathcal X}(\mathcal X)d \mathcal XPX(X)dX必然相等;但是在定积分中,ZK,X\mathcal Z_{\mathcal K},\mathcal XZK,X位于不同的特征空间对应的积分值(有正有负)存在差异。因此需要加上‘模’符号。 ∣PZK(ZK)dZK∣∣PX(X)dX∣|\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K}) d\mathcal Z_{\mathcal K}| |P_{\mathcal X}(\mathcal X) d\mathcal X|∣PZK(ZK)dZK∣∣PX(X)dX∣ 但由于PZK(ZK),PX(X)\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K}),\mathcal P_{\mathcal X}(\mathcal X)PZK(ZK),PX(X)它们是概率密度函数它们的实际结果表示概率值(恒正)。因此∣PX(X)∣PX(X)|\mathcal P_{\mathcal X}(\mathcal X)| \mathcal P_{\mathcal X}(\mathcal X)∣PX(X)∣PX(X)PZK(ZK)\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PZK(ZK)同理。经过移项可将概率分布之间的关系表示为如下形式 PX(X)∣dZKdX∣⋅PZK(ZK)\mathcal P_{\mathcal X}(\mathcal X) \left|\frac{d\mathcal Z_{\mathcal K}}{d\mathcal X}\right| \cdot \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PX(X)dXdZK⋅PZK(ZK) 将ZKfK−1(X)\mathcal Z_{\mathcal K} f_{\mathcal K}^{-1}(\mathcal X)ZKfK−1(X)代入最终可得到如下形式 PX(X)∣∂fK−1(X)∂X∣⋅PZK(ZK)\mathcal P_{\mathcal X}(\mathcal X) \left|\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right| \cdot \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PX(X)∂X∂fK−1(X)⋅PZK(ZK)观察系数项∣∂fK−1(X)∂X∣\left|\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right|∂X∂fK−1(X)它是一个标量、常数但∂fK−1(X)∂X\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial\mathcal X}∂X∂fK−1(X)自身是一个矩阵 该矩阵被称作雅可比矩阵Jacobian\text{Jacobian}Jacobian ∂fK−1(X)∂X[∂fK−1(X1)∂X1∂fK−1(X1)∂X2⋯∂fK−1(X1)∂Xp∂fK−1(X2)∂X1∂fK−1(X2)∂X2⋯∂fK−1(X2)∂Xp⋮⋮⋱⋮∂fK−1(Xp)∂X1∂fK−1(Xp)∂X2⋯∂fK−1(Xp)∂Xp]p×p\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X} \begin{bmatrix} \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_1)}{\partial \mathcal X_1} \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_1)}{\partial \mathcal X_2} \cdots \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_1)}{\partial \mathcal X_p} \\ \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_2)}{\partial \mathcal X_1} \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_2)}{\partial \mathcal X_2} \cdots \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_2)}{\partial \mathcal X_p}\\ \vdots \vdots \ddots \vdots\\ \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_p)}{\partial \mathcal X_1} \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_p)}{\partial \mathcal X_2} \cdots \frac{\partial f_{\mathcal K}^{-1}(\mathcal X_p)}{\partial \mathcal X_p} \end{bmatrix}_{p \times p}∂X∂fK−1(X)∂X1∂fK−1(X1)∂X1∂fK−1(X2)⋮∂X1∂fK−1(Xp)∂X2∂fK−1(X1)∂X2∂fK−1(X2)⋮∂X2∂fK−1(Xp)⋯⋯⋱⋯∂Xp∂fK−1(X1)∂Xp∂fK−1(X2)⋮∂Xp∂fK−1(Xp)p×p 那么∣∂fK−1(X)∂X∣\left|\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right|∂X∂fK−1(X)实际上是与雅克比矩阵对应的雅克比行列式(Jacobian Determinant\text{Jacobian Determinant}Jacobian Determinant)的绝对值。使用det[∂fK−1(X)∂X]\text{det}\left[\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right]det[∂X∂fK−1(X)]进行表示 PX(X)∣det[∂fK−1(X)∂X]∣⋅PZK(ZK)\mathcal P_{\mathcal X}(\mathcal X) \left|\text{det}\left[\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right]\right| \cdot \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PX(X)det[∂X∂fK−1(X)]⋅PZK(ZK)继续变换观察∂fK−1(X)∂X\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}∂X∂fK−1(X)可以继续向下变换 {∂fK−1(X)∂X⋅∂fK(ZK)∂ZK1⇒∂fK−1(X)∂X[∂fK(ZK)∂ZK]−1⇒∣det[∂fK−1(X)∂X]∣∣det[∂fK(ZK)∂ZK]∣−1\begin{cases} \frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X} \cdot \frac{\partial f_{\mathcal K}(\mathcal Z_{\mathcal K})}{\partial \mathcal Z_{\mathcal K}} 1 \Rightarrow \frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X} \left[\frac{\partial f_{\mathcal K}(\mathcal Z_{\mathcal K})}{\partial \mathcal Z_{\mathcal K}}\right]^{-1} \\ \Rightarrow \left|\text{det}\left[\frac{\partial f_{\mathcal K}^{-1}(\mathcal X)}{\partial \mathcal X}\right]\right| \left|\text{det}\left[\frac{\partial f_{\mathcal K}(\mathcal Z_{\mathcal K})}{\partial \mathcal Z_{\mathcal K}}\right]\right|^{-1} \end{cases}⎩⎨⎧∂X∂fK−1(X)⋅∂ZK∂fK(ZK)1⇒∂X∂fK−1(X)[∂ZK∂fK(ZK)]−1⇒det[∂X∂fK−1(X)]det[∂ZK∂fK(ZK)]−1 最终分布PX(X)\mathcal P_{\mathcal X}(\mathcal X)PX(X)与分布PZK(ZK)\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PZK(ZK)之间的关系表示为 PX(X)∣det[∂fK(ZK)∂ZK]∣−1⋅PZK(ZK)\mathcal P_{\mathcal X}(\mathcal X) \left|\text{det}\left[\frac{\partial f_{\mathcal K}(\mathcal Z_{\mathcal K})}{\partial \mathcal Z_{\mathcal K}}\right]\right|^{-1} \cdot \mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PX(X)det[∂ZK∂fK(ZK)]−1⋅PZK(ZK)
至此从随机变量ZK\mathcal Z_{\mathcal K}ZK与随机变量X\mathcal XX之间的函数关系转化为概率分布PX(X)\mathcal P_{\mathcal X}(\mathcal X)PX(X)与PZK(ZK)\mathcal P_{\mathcal Z_{\mathcal K}}(\mathcal Z_{\mathcal K})PZK(ZK)之间的函数关系已表示出来。而流模型中的每一个过程均是基于上述关系一层一层计算过来。
不同于以往对P(X)\mathcal P(\mathcal X)P(X)的求解过程它能够将P(X)\mathcal P(\mathcal X)P(X)描述出来直到使用隐变量的层数选择完成其对应的P(X)\mathcal P(\mathcal X)P(X)计算精度达到条件即可。关于流模型的学习方式依然是极大似然估计(Maximum Likelihood Estimation,MLE\text{Maximum Likelihood Estimation,MLE}Maximum Likelihood Estimation,MLE) logPX(X)log{∏k1K∣det[∂fk(Zk)∂Zk]∣−1⋅Pinit(Zinit)}logPinit(Zinit)∑k1Klog{∣det[∂fk(Zk)∂Zk]∣−1}\begin{aligned} \log \mathcal P_{\mathcal X}(\mathcal X) \log \left\{\prod_{k1}^{\mathcal K} \left|\text{det} \left[\frac{\partial f_{k}(\mathcal Z_k)}{\partial \mathcal Z_k}\right]\right|^{-1} \cdot \mathcal P_{init}(\mathcal Z_{init})\right\} \\ \log \mathcal P_{init}(\mathcal Z_{init}) \sum_{k1}^{\mathcal K} \log \left\{\left|\text{det} \left[\frac{\partial f_{k}(\mathcal Z_k)}{\partial \mathcal Z_k}\right]\right|^{-1}\right\} \end{aligned}logPX(X)log{k1∏Kdet[∂Zk∂fk(Zk)]−1⋅Pinit(Zinit)}logPinit(Zinit)k1∑Klog{det[∂Zk∂fk(Zk)]−1}
相关参考 雅可比矩阵——百度百科 【机器学习白板推导系列(三十三) ~ 流模型(Flow Based Model)】
