深度学习中的矩阵微积分详解

需积分: 32 31 浏览量更新于2024-07-17 收藏 743KB PDF 举报

"这篇文档是关于深度学习中矩阵微积分基础知识的介绍，旨在帮助读者理解深度神经网络的训练过程。作者Terence Parr和Jeremy Howard在数据科学领域有丰富的教学和项目经验。文档提供了对初等微积分知识的要求，并在需要时提供数学复习链接，适合已经了解神经网络基础并希望深入理解其数学原理的读者。" 深度学习是一种强大的机器学习技术，尤其在图像识别、自然语言处理等领域表现出色。在这个过程中，理解和应用矩阵微积分是至关重要的，因为它是优化模型（如深度神经网络）的关键。这篇文档深入浅出地讲解了矩阵微积分在深度学习中的应用，帮助那些已经具备一定深度学习实践经验但想进一步提升理论知识的人。矩阵微积分是微积分的扩展，它处理多变量函数，特别适用于处理高维数据，比如神经网络中权重矩阵和激活函数的计算。在深度学习中，矩阵微积分主要体现在以下几个方面： 1. **梯度计算**：在训练神经网络时，我们需要计算损失函数关于权重参数的梯度，这是通过反向传播算法实现的。矩阵微积分提供了一种计算这些梯度的方法，如链式法则和雅可比矩阵。 2. **优化算法**：优化算法如梯度下降法和其变种（如动量法、Adam等）依赖于梯度信息来更新网络权重。这些更新通常涉及矩阵乘法和偏导运算。 3. **正则化**：正则化如L1和L2正则化也是通过矩阵微积分来实施的，它有助于防止过拟合，保持模型的泛化能力。 4. **批量归一化**：批量归一化的操作涉及到输入数据矩阵的统计属性计算，如均值和方差，这需要用到矩阵微积分。 5. **损失函数**：深度学习中的损失函数，如交叉熵损失，通常是对整个批次样本的损失的平均，这需要对矩阵进行操作。 6. **卷积运算**：在卷积神经网络中，卷积操作本质上是矩阵乘法，因此矩阵微积分在这里同样重要。 7. **张量运算**：深度学习框架如TensorFlow和PyTorch用到大量的张量运算，这些运算本质上都是矩阵运算的扩展，微积分概念在其中起着核心作用。该文档将帮助读者理解这些概念，通过实例和解释，让读者能够独立计算和分析深度学习模型的性能。虽然不必在实际使用深度学习前完全掌握这些知识，但深入理解矩阵微积分可以帮助我们更好地设计和调试模型，从而提升模型的性能。如果在阅读过程中遇到困难，可以回溯至前面的章节或利用提供的数学复习资源加深理解。

with respect to x is written

∂

∂x

y. There are three constants from the perspective of

∂

∂x

: 3, 2,

and y. Therefore,

∂

∂x

3yx

= 3y

∂

∂x

= 3y2x = 6yx. The partial derivative with respect to y treats

x like a constant:

∂

∂y

y = 3x

∂

∂y

y = 3x

∂y

= 3x

× 1 = 3x

. It’s a good idea to derive these

yourself before continuing otherwise the rest of the article won’t make sense. Here’s the Khan

Academy video on partials if you need help.

To make it clear we are doing vector calculus and not just multivariate calculus, let’s consider what

we do with the partial derivatives

∂f (x,y)

∂x

and

∂f (x,y)

∂y

(another way to say

∂

∂x

f(x, y) and

∂

∂y

f(x, y))

that we computed for f(x, y) = 3x

y. Instead of having them just ﬂoating around and not organized

in any way, let’s organize them into a horizontal vector. We call this vector the gradient of f(x, y)

and write it as:

∇f(x, y) = [

∂f(x, y)

∂x

∂f(x, y)

∂y

] = [6yx, 3x

]

So the gradient of f(x, y) is simply a vector of its partials. Gradients are part of the vector calculus

world, which deals with functions that map n scalar parameters to a single scalar. Now, let’s get

crazy and consider derivatives of multiple functions simultaneously.

4 Matrix calculus

When we move from derivatives of one function to derivatives of many functions, we move from

the world of vector calculus to matrix calculus. Let’s compute partial derivatives for two functions,

both of which take two parameters. We can keep the same f(x, y) = 3x

y from the last section,

but let’s also bring in g(x, y) = 2x + y

. The gradient for g has two entries, a partial derivative for

each parameter:

∂g(x, y)

∂x

∂2x

∂x

∂y

∂x

= 2

∂x

+ 0 = 2 × 1 = 2

and

∂g(x, y)

∂y

∂2x

∂y

= 0 + 8y

= 8y

giving us gradient ∇g(x, y) = [2, 8y

Gradient vectors organize all of the partial derivatives for a speciﬁc scalar function. If we have two

functions, we can also organize their gradients into a matrix by stacking the gradients. When we

do so, we get the Jacobian matrix (or just the Jacobian) where the gradients are rows:

J =



∇f(x, y)

∇g(x, y)



∂f (x,y)

∂x

∂f (x,y)

∂y

∂g(x,y)

∂x

∂g(x,y)

∂y



6yx 3x

2 8y



Welcome to matrix calculus!

Note that there are multiple ways to represent the Jacobian. We are using the so-called

numerator layout but many papers and software will use the denominator layout. This is just

transpose of the numerator layout Jacobian (ﬂip it around its diagonal):



6yx 2



4.1 Generalization of the Jacobian

So far, we’ve looked at a speciﬁc example of a Jacobian matrix. To deﬁne the Jacobian matrix

more generally, let’s combine multiple parameters into a single vector argument: f(x, y, z) ⇒ f(x).

(You will sometimes see notation ~x for vectors in the literature as well.) Lowercase letters in bold

font such as x are vectors and those in italics font like x are scalars. x

is the i

element of vector

x and is in italics because a single vector element is a scalar. We also have to deﬁne an orientation

for vector x. We’ll assume that all vectors are vertical by default of size n × 1:

x =













With multiple scalar-valued functions, we can combine them all into a vector just like we did with

the parameters. Let y = f(x) be a vector of m scalar-valued functions that each take a vector x

of length n = |x| where |x| is the cardinality (count) of elements in x. Each f

function within f

returns a scalar just as in the previous section:

= f

(x)

= f

(x)

= f

(x)

For instance, we’d represent f (x, y) = 3x

y and g(x, y) = 2x + y

from the last section as

= f

(x) = 3x

(substituting x

for x, x

for y)

= f

(x) = 2x

+ x

It’s very often the case that m = n because we will have a scalar function result for each element

of the x vector. For example, consider the identity function y = f(x) = x:

= f

(x) = x

= f

(x) = x

= f

(x) = x

So we have m = n functions and parameters, in this case. Generally speaking, though, the Jacobian

matrix is the collection of all m × n possible partial derivatives (m rows and n columns), which is

the stack of m gradients with respect to x:

∂y

∂x







∇f

(x)

∇f

(x)

. . .

∇f

(x)













∂

∂x

(x)

∂

∂x

(x)

. . .

∂

∂x

(x)













∂

∂x

(x)

∂

∂x

(x) . . .

∂

∂x

(x)

∂

∂x

(x)

∂

∂x

(x) . . .

∂

∂x

(x)

. . .

∂

∂x

(x)

∂

∂x

(x) . . .

∂

∂x

(x)







剩余32页未读，继续阅读

CCCCCCCCCCCCCCC

粉丝: 73
资源: 96

深度学习中的矩阵微积分详解

《神经网络与深度学习》- 矩阵微积分简介

深度学习中的矩阵微积分实践指南

深度学习入门：数学基础与向量矩阵解析

深度学习之微积分基础.zip

深度学习资料：矩阵微积分、机器学习的数学基础、CS229线性代数与概率论基础和机器学习、深度学习和自然语言处理一整套资源

The Matrix Calculus You Need For Deep Learning 《深度学习所需的矩阵微积分知识》

深度学习基础及数学原理.zip_深度_深度学习_深度学习原理_深度学习数学原理_深度学习算法

深度学习系统笔记，包含深度学习数学基础知识、神经网络基础部件详解、深度学习炼丹策略、模型压缩算法详解等.zip

深度学习基石：微积分与线性代数精讲

矩阵微积分：统计与经济计量学中的应用（第三版）

最新资源