\exp

chapter_appendix-mathematics-for-deep-learning/distributions.md

@@ -752,7 +752,7 @@ belong to which is known as the *exponential family*. The exponential family
 is a set of distributions whose density can be expressed in the following 
 form:
 
-$$p(\mathbf{x} \mid \boldsymbol{\eta}) = h(\mathbf{x}) \cdot \textrm{exp} \left( \boldsymbol{\eta}^{\top} \cdot T(\mathbf{x}) - A(\boldsymbol{\eta}) \right)$$
+$$p(\mathbf{x} \mid \boldsymbol{\eta}) = h(\mathbf{x}) \cdot \exp \left( \boldsymbol{\eta}^{\top} \cdot T(\mathbf{x}) - A(\boldsymbol{\eta}) \right)$$
 :eqlabel:`eq_exp_pdf`
 
 As this definition can be a little subtle, let's examine it closely.  
@@ -777,17 +777,17 @@ Third, we have $A(\boldsymbol{\eta})$, which is referred to as the *cumulant
 function*, which ensures that the above distribution :eqref:`eq_exp_pdf` 
 integrates to one, i.e.,
 
-$$A(\boldsymbol{\eta})  = \log \left[\int h(\mathbf{x}) \cdot \textrm{exp}
+$$A(\boldsymbol{\eta})  = \log \left[\int h(\mathbf{x}) \cdot \exp
 \left(\boldsymbol{\eta}^{\top} \cdot T(\mathbf{x}) \right) d\mathbf{x} \right].$$
 
 To be concrete, let's consider the Gaussian. Assuming that $\mathbf{x}$ is 
 an univariate variable, we saw that it had a density of
 
 $$
 \begin{aligned}
-p(x \mid \mu, \sigma) &= \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \textrm{exp} 
+p(x \mid \mu, \sigma) &= \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \exp 
 \left\{ \frac{-(x-\mu)^2}{2 \sigma^2} \right\} \\
-&= \frac{1}{\sqrt{2 \pi}} \cdot \textrm{exp} \left\{ \frac{\mu}{\sigma^2}x
+&= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left\{ \frac{\mu}{\sigma^2}x
 -\frac{1}{2 \sigma^2} x^2 - \left( \frac{1}{2 \sigma^2} \mu^2
 +\log(\sigma) \right) \right\}.
 \end{aligned}

chapter_natural-language-processing-pretraining/glove.md

@@ -37,7 +37,7 @@ of word $w_j$ given word $w_i$
 in the skip-gram model,
 we have
 
-$$q_{ij}=\frac{\exp(\mathbf{u}_j^\top \mathbf{v}_i)}{ \sum_{k \in \mathcal{V}} \textrm{exp}(\mathbf{u}_k^\top \mathbf{v}_i)},$$
+$$q_{ij}=\frac{\exp(\mathbf{u}_j^\top \mathbf{v}_i)}{ \sum_{k \in \mathcal{V}} \exp(\mathbf{u}_k^\top \mathbf{v}_i)},$$
 
 where
 for any index $i$

chapter_natural-language-processing-pretraining/word2vec.md

@@ -98,7 +98,7 @@ context word $w_o$ (with index $o$ in the dictionary) given the center word $w_c
 a softmax operation on vector dot products:
 
 
-$$P(w_o \mid w_c) = \frac{\textrm{exp}(\mathbf{u}_o^\top \mathbf{v}_c)}{ \sum_{i \in \mathcal{V}} \textrm{exp}(\mathbf{u}_i^\top \mathbf{v}_c)},$$
+$$P(w_o \mid w_c) = \frac{\exp(\mathbf{u}_o^\top \mathbf{v}_c)}{ \sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \mathbf{v}_c)},$$
 :eqlabel:`eq_skip-gram-softmax`
 
 where the vocabulary index set $\mathcal{V} = \{0, 1, \ldots, |\mathcal{V}|-1\}$.
@@ -137,13 +137,13 @@ involving any pair of the center word $w_c$ and
 the context word $w_o$ is
 
 
-$$\log P(w_o \mid w_c) =\mathbf{u}_o^\top \mathbf{v}_c - \log\left(\sum_{i \in \mathcal{V}} \textrm{exp}(\mathbf{u}_i^\top \mathbf{v}_c)\right).$$
+$$\log P(w_o \mid w_c) =\mathbf{u}_o^\top \mathbf{v}_c - \log\left(\sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \mathbf{v}_c)\right).$$
 :eqlabel:`eq_skip-gram-log`
 
 Through differentiation, we can obtain its gradient
 with respect to the center word vector $\mathbf{v}_c$ as
 
-$$\begin{aligned}\frac{\partial \textrm{log}\, P(w_o \mid w_c)}{\partial \mathbf{v}_c}&= \mathbf{u}_o - \frac{\sum_{j \in \mathcal{V}} \exp(\mathbf{u}_j^\top \mathbf{v}_c)\mathbf{u}_j}{\sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \mathbf{v}_c)}\\&= \mathbf{u}_o - \sum_{j \in \mathcal{V}} \left(\frac{\textrm{exp}(\mathbf{u}_j^\top \mathbf{v}_c)}{ \sum_{i \in \mathcal{V}} \textrm{exp}(\mathbf{u}_i^\top \mathbf{v}_c)}\right) \mathbf{u}_j\\&= \mathbf{u}_o - \sum_{j \in \mathcal{V}} P(w_j \mid w_c) \mathbf{u}_j.\end{aligned}$$
+$$\begin{aligned}\frac{\partial \textrm{log}\, P(w_o \mid w_c)}{\partial \mathbf{v}_c}&= \mathbf{u}_o - \frac{\sum_{j \in \mathcal{V}} \exp(\mathbf{u}_j^\top \mathbf{v}_c)\mathbf{u}_j}{\sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \mathbf{v}_c)}\\&= \mathbf{u}_o - \sum_{j \in \mathcal{V}} \left(\frac{\exp(\mathbf{u}_j^\top \mathbf{v}_c)}{ \sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \mathbf{v}_c)}\right) \mathbf{u}_j\\&= \mathbf{u}_o - \sum_{j \in \mathcal{V}} P(w_j \mid w_c) \mathbf{u}_j.\end{aligned}$$
 :eqlabel:`eq_skip-gram-grad`
 
 
@@ -194,7 +194,7 @@ center word $w_c$ (with index $c$ in the dictionary) given its surrounding conte
 
 
 
-$$P(w_c \mid w_{o_1}, \ldots, w_{o_{2m}}) = \frac{\textrm{exp}\left(\frac{1}{2m}\mathbf{u}_c^\top (\mathbf{v}_{o_1} + \ldots + \mathbf{v}_{o_{2m}}) \right)}{ \sum_{i \in \mathcal{V}} \textrm{exp}\left(\frac{1}{2m}\mathbf{u}_i^\top (\mathbf{v}_{o_1} + \ldots + \mathbf{v}_{o_{2m}}) \right)}.$$
+$$P(w_c \mid w_{o_1}, \ldots, w_{o_{2m}}) = \frac{\exp\left(\frac{1}{2m}\mathbf{u}_c^\top (\mathbf{v}_{o_1} + \ldots + \mathbf{v}_{o_{2m}}) \right)}{ \sum_{i \in \mathcal{V}} \exp\left(\frac{1}{2m}\mathbf{u}_i^\top (\mathbf{v}_{o_1} + \ldots + \mathbf{v}_{o_{2m}}) \right)}.$$
 :eqlabel:`fig_cbow-full`
 
 
@@ -232,7 +232,7 @@ with respect to any context word vector $\mathbf{v}_{o_i}$($i = 1, \ldots, 2m$)
 as
 
 
-$$\frac{\partial \log\, P(w_c \mid \mathcal{W}_o)}{\partial \mathbf{v}_{o_i}} = \frac{1}{2m} \left(\mathbf{u}_c - \sum_{j \in \mathcal{V}} \frac{\exp(\mathbf{u}_j^\top \bar{\mathbf{v}}_o)\mathbf{u}_j}{ \sum_{i \in \mathcal{V}} \textrm{exp}(\mathbf{u}_i^\top \bar{\mathbf{v}}_o)} \right) = \frac{1}{2m}\left(\mathbf{u}_c - \sum_{j \in \mathcal{V}} P(w_j \mid \mathcal{W}_o) \mathbf{u}_j \right).$$
+$$\frac{\partial \log\, P(w_c \mid \mathcal{W}_o)}{\partial \mathbf{v}_{o_i}} = \frac{1}{2m} \left(\mathbf{u}_c - \sum_{j \in \mathcal{V}} \frac{\exp(\mathbf{u}_j^\top \bar{\mathbf{v}}_o)\mathbf{u}_j}{ \sum_{i \in \mathcal{V}} \exp(\mathbf{u}_i^\top \bar{\mathbf{v}}_o)} \right) = \frac{1}{2m}\left(\mathbf{u}_c - \sum_{j \in \mathcal{V}} P(w_j \mid \mathcal{W}_o) \mathbf{u}_j \right).$$
 :eqlabel:`eq_cbow-gradient`