python - 如何在keras中实现麦克劳林系列？

Question

我正在尝试使用 maclaurin 系列来实现可扩展的 CNN。基本思想是可以将第一个输入节点分解为具有不同阶数和系数的多个节点。将单个节点分解为多个节点可以生成与麦克劳林级数不同的非线性线连接。谁能给我一个关于如何扩展的可能想法 CNN与麦克劳林级数非线性展开？任何想法？

我不太明白如何将输入节点分解为多个具有不同非线性线连接的节点，这些节点由麦克劳林级数生成。据我所知，麦克劳林级数是一个近似函数，但分解节点在实现方面对我来说不是很直观。如何在python中实现一个分解输入节点到多个节点？如何轻松实现这一目标？任何的想法？

我的尝试 :

import tensorflow as tf
import numpy as np
import keras
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense, Conv2D, MaxPooling2D, Dropout, Flatten
from keras.datasets import cifar10
from keras.utils import to_categorical

(train_imgs, train_label), (test_imgs, test_label)= cifar10.load_data()
output_class = np.unique(train_label)
n_class = len(output_class)

nrows_tr, ncols_tr, ndims_tr = train_imgs.shape[1:]
nrows_ts, ncols_ts, ndims_ts = test_imgs.shape[1:]
train_data = train_imgs.reshape(train_imgs.shape[0], nrows_tr, ncols_tr, ndims_tr)

test_data = test_imgs.reshape(test_imgs.shape[0], nrows_ts, ncols_ts, ndims_ts)
input_shape = (nrows_tr, ncols_tr, ndims_tr)
train_data = train_data.astype('float32')
trast_data = test_data.astype('float32')
train_data //= 255
test_data //= 255
train_label_one_hot = to_categorical(train_label)
test_label_one_hot = to_categorical(test_label)

def pown(x,n):
    return(x**n)

def expandable_cnn(input_shape, output_shape, approx_order):
    inputs=Input(shape=(input_shape))
    x= Dense(input_shape)(inputs)
    y= Dense(output_shape)(x)
    model = Sequential()
    model.add(Conv2D(filters=32, kernel_size=(3,3), padding='same', activation="relu", input_shape=input_shape))
    model.add(Conv2D(filters=32, kernel_size=(3,3), activation='relu'))
    model.add(MaxPooling2D(pool_size=(2,2)))
    model.add(Dropout(0.25))

    model.add(Flatten())
    model.add(Dense(512, activation='relu'))
    model.add(Dropout(0.5))
    for i in range(2, approx_order+1):
        y=add([y, Dense(output_shape)(Activation(lambda x: pown(x, n=i))(x))])
    model.add(Dense(n_class, activation='softmax')(y))
    return model

但是当我运行上面的模型时，我有一堆编译错误和维度错误。我认为 CNN 模型的 Tylor 非线性扩展方式可能不正确。另外，我不确定如何表示重量。如何使这项工作？关于如何纠正我的尝试的任何可能的想法？

所需的输出 :

我期待用麦克劳林级数非线性展开来扩展CNN，如何使上述实现正确有效？任何可能的想法或方法？

Answer 1

有趣的问题。我已经实现了一个 Keras 模型，它按照您的描述计算泰勒展开:

from tensorflow.keras.models import Model
from tensorflow.keras.layers import Dense, Input, Lambda


def taylor_expansion_network(input_dim, max_pow):
    x = Input((input_dim,))

    # 1. Raise input x_i to power p_i for each i in [0, max_pow].
    def raise_power(x, max_pow):
        x_ = x[..., None]  # Shape=(batch_size, input_dim, 1)
        x_ = tf.tile(x_, multiples=[1, 1, max_pow + 1])  # Shape=(batch_size, input_dim, max_pow+1)
        pows = tf.range(0, max_pow + 1, dtype=tf.float32)  # Shape=(max_pow+1,)
        x_p = tf.pow(x_, pows)  # Shape=(batch_size, input_dim, max_pow+1)
        x_p_ = x_p[..., None]  # Shape=(batch_size, input_dim, max_pow+1, 1)
        return x_p_

    x_p_ = Lambda(lambda x: raise_power(x, max_pow))(x)

    # 2. Multiply by alpha coefficients
    h = LocallyConnected2D(filters=1,
                           kernel_size=1,  # This layer is computing a_i * x^{p_i} for each i in [0, max_pow]
                           use_bias=False)(x_p_)  # Shape=(batch_size, input_dim, max_pow+1, 1)

    # 3. Compute s_i for each i in [0, max_pow]
    def cumulative_sum(h):
        h = tf.squeeze(h, axis=-1)  # Shape=(batch_size, input_dim, max_pow+1)
        s = tf.cumsum(h, axis=-1)  # s_i = sum_{j=0}^i h_j. Shape=(batch_size, input_dim, max_pow+1)
        s_ = s[..., None]  # Shape=(batch_size, input_dim, max_pow+1, 1)
        return s_

    s_ = Lambda(cumulative_sum)(h)

    # 4. Compute sum w_i * s_i each i in [0, max_pow]
    s_ = LocallyConnected2D(filters=1,  # This layer is computing w_i * s_i for each i in [0, max_pow]
                            kernel_size=1,
                            use_bias=False)(s_)  # Shape=(batch_size, input_dim, max_pow+1)
    y = Lambda(lambda s_: tf.reduce_sum(tf.squeeze(s_, axis=-1), axis=-1))(s_)  # Shape=(batch_size, input_dim)

    # Return Taylor expansion model
    model = Model(inputs=x, outputs=y)
    model.summary()
    return model

该实现将相同的泰勒展开式应用于形状为 (batch_size, input_dim=512) 的扁平张量的每个元素。来自卷积网络。

更新 :正如我们在评论部分所讨论的，这里有一些代码来展示你的函数 expandable_cnn可以修改以集成上面定义的模型:

def expandable_cnn(input_shape, nclass, approx_order):
    inputs = Input(shape=(input_shape))
    h = inputs
    h = Conv2D(filters=32, kernel_size=(3, 3), padding='same', activation='relu', input_shape=input_shape)(h)
    h = Conv2D(filters=32, kernel_size=(3, 3), activation='relu')(h)
    h = MaxPooling2D(pool_size=(2, 2))(h)
    h = Dropout(0.25)(h)
    h = Flatten()(h)
    h = Dense(512, activation='relu')(h)
    h = Dropout(0.5)(h)
    taylor_model = taylor_expansion_network(input_dim=512, max_pow=approx_order)
    h = taylor_model(h)
    h = Activation('relu')(h)
    print(h.shape)
    h = Dense(nclass, activation='softmax')(h)
    model = Model(inputs=inputs, outputs=h)
    return model

请注意，我不保证您的模型会起作用(例如，您将获得良好的性能)。我只是根据我对您想要的内容的解释提供了一个解决方案。