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279 lines
5.7 KiB
Markdown
279 lines
5.7 KiB
Markdown
# Theano tensor 模块：操作符和逐元素操作
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## 操作符
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In [1]:
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```py
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import theano
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from theano import tensor as T
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```
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```py
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Using gpu device 1: Tesla C2075 (CNMeM is disabled)
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```
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`tensor` 类型支持很多基本的操作：
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In [2]:
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```py
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# 两个整形三维张量
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a, b = T.itensor3("a"), T.itensor3("b")
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```
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### 算术操作
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In [3]:
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```py
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print theano.pp(a + 3) # T.add(a, 3) -> itensor3
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print theano.pp(3 - a) # T.sub(3, a)
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print theano.pp(a * 3.5) # T.mul(a, 3.5) -> ftensor3 or dtensor3 (depending on casting)
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print theano.pp(2.2 / a) # T.truediv(2.2, a)
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print theano.pp(2.2 // a) # T.intdiv(2.2, a)
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print theano.pp(2.2**a) # T.pow(2.2, a)
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print theano.pp(b % a) # T.mod(b, a)
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```
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```py
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(a + TensorConstant{3})
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(TensorConstant{3} - a)
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(a * TensorConstant{3.5})
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(TensorConstant{2.20000004768} / a)
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(TensorConstant{2.20000004768} // a)
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(TensorConstant{2.20000004768} ** a)
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mod(b, a)
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```
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### 比特操作
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In [4]:
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```py
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print theano.pp(a & b) # T.and_(a,b) bitwise and (alias T.bitwise_and)
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print theano.pp(a ^ 1) # T.xor(a,1) bitwise xor (alias T.bitwise_xor)
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print theano.pp(a | b) # T.or_(a,b) bitwise or (alias T.bitwise_or)
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print theano.pp(~a) # T.invert(a) bitwise invert (alias T.bitwise_not)
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```
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```py
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and_(a, b)
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xor(a, TensorConstant{1})
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or_(a, b)
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invert(a)
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```
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### 原地操作
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`Theano` 不支持原地操作如 `+=` 等，`Theano` 的图优化解构会自动决定是否使用原地操作。如果需要更新变量的值，可以考虑使用共享变量 `theano.shared`。
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## 逐元素操作
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### 类型转换
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`T.cast(x, dtype)` 用于类型转换：
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In [5]:
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```py
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x = T.matrix()
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x_as_int = T.cast(x, 'int32')
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```
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`T.cast(x, dtype)` 的机制与 `numpy.asarray(x, dtype)` 的机制类似，只有 `dtype` 不同时才会创建新的变量：
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In [6]:
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```py
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print x_as_int is x
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print T.cast(x, theano.config.floatX) is x
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```
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```py
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False
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True
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```
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复数取实部，虚部，角度，模：
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* `T.real(a)`
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* `T.imag(a)`
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* `T.angle(a)`
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* `T.abs_(a)`
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### 比较
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`Theano` 的比较操作也是逐元素的：
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* `T.lt(a, b)` : <
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* `T.gt(a, b)` : >
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* `T.le(a, b)` : <=
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* `T.ge(a, b)` : >=
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* `T.eq(a, b)` : ==
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* `T.neq(a, b)` : !=
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`Theano` 中没有 `bool` 类型，所有的 `bool` 类型都用 `int8` 表示。
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In [7]:
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```py
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x, y = T.dmatrices('x','y')
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print theano.pp(T.le(x, y))
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```
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```py
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le(x, y)
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```
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除此之外，还有另一些与 `numpy` 类似的用法：
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* `T.isnan(a)` : 是否 NAN
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* `T.isinf(a)` : 是否 INF
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* `T.isclose(a, b)` ：浮点数是否接近
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* `T.allclose(a, b)` ：浮点数是否很接近
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### 条件
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`T.switch(cond, ift, iff)` 选择 `ift (if ture)` 和 `iff (if false)`。
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`T.where(cond, ift, iff)` 与 `switch` 一致。
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`T.clip(x, min, max)` 低于 `min` 的部分变成 `min`，超过 `max` 的部分变成 `max`。
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### 数学操作
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In [8]:
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```py
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a, b = T.matrices("a", "b")
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print theano.pp(T.maximum(a, b)) # max(a, b)
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print theano.pp(T.minimum(a, b)) # min(a, b)
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print theano.pp(T.neg(a)) # -a
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print theano.pp(T.inv(a)) # 1.0/a
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print theano.pp(T.exp(a))
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print theano.pp(T.log(a)), theano.pp(T.log2(a)), theano.pp(T.log10(a)) # log10(a)
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print theano.pp(T.sgn(a)) # sgn(a)
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print theano.pp(T.floor(a)) # floor(a)
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print theano.pp(T.ceil(a)) # ceil(a)
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print theano.pp(T.round(a)) # round(a)
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print theano.pp(T.iround(a)) # iround(a)
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print theano.pp(T.sqr(a)) # sqr(a)
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print theano.pp(T.sqrt(a)) # sqrt(a)
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print theano.pp(T.cos(a)), theano.pp(T.sin(a)), theano.pp(T.tan(a))
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print theano.pp(T.cosh(a)), theano.pp(T.sinh(a)), theano.pp(T.tanh(a)) # tan(a)
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print theano.pp(T.erf(a)), theano.pp(T.erfc(a)) # erf(a), erfc(a)
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print theano.pp(T.erfinv(a)), theano.pp(T.erfcinv(a))
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print theano.pp(T.gamma(a)) # gamma(a)
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print theano.pp(T.gammaln(a)) # log(gamma(a))
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print theano.pp(T.psi(a)) # digamma(a)
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```
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```py
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maximum(a, b)
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minimum(a, b)
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(-a)
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inv(a)
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exp(a)
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log(a) log2(a) log10(a)
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sgn(a)
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floor(a)
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ceil(a)
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round_half_away_from_zero(a)
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int64(round_half_away_from_zero(a))
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sqr(a)
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sqrt(a)
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cos(a) sin(a) tan(a)
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cosh(a) sinh(a) tanh(a)
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erf(a) erfc(a)
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erfinv(a) erfcinv(a)
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gamma(a)
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gammaln(a)
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psi(a)
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```
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其中 `erf, erfc` 定义如下： [https://en.wikipedia.org/wiki/Error_function](https://en.wikipedia.org/wiki/Error_function)
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$$ \operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_0^x e^{-t^2} dt $$$$ \begin{align} \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\,\mathrm dt \\ & = e^{-x^2} \operatorname{erfcx}(x) \end{align} $$
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`erfinv, erfcinv` 为其反函数：1 [https://en.wikipedia.org/wiki/Error_function#Inverse_functions](https://en.wikipedia.org/wiki/Error_function#Inverse_functions)
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### Broadcasting
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图示如上。
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## 线性代数
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矩阵乘法：`T.dot(x, y)`
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向量外积：`T.outer(x, y)`
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张量乘法：`tensordot(a, b, axes=2)`
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`axes` 参数表示 `a` `b` 对应要去掉的维度。
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In [9]:
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```py
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import numpy as np
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a = np.random.random((2,3,4))
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b = np.random.random((5,6,4,3))
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#tensordot
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c = np.tensordot(a, b, [[1,2],[3,2]])
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#loop replicating tensordot
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a0, a1, a2 = a.shape
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b0, b1, _, _ = b.shape
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cloop = np.zeros((a0,b0,b1))
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#loop over non-summed indices -- these exist
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#in the tensor product.
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for i in range(a0):
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for j in range(b0):
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for k in range(b1):
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#loop over summed indices -- these don't exist
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#in the tensor product.
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for l in range(a1):
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for m in range(a2):
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cloop[i,j,k] += a[i,l,m] * b[j,k,m,l]
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assert np.allclose(c, cloop)
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print a.shape, b.shape
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print c.shape
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```
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```py
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(2, 3, 4) (5, 6, 4, 3)
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(2, 5, 6)
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``` |