如何在keras中实现麦克劳林级数？

Question

我正在尝试使用麦克劳林级数来实现可扩展的 CNN。基本思想是第一个输入节点可以分解为多个具有不同阶数和系数的节点。将单个节点分解为多个节点可以生成由麦克劳林级数生成的不同非线性线连接。谁能告诉我如何用麦克劳林级数非线性展开来展开 CNN？有什么想法吗？

我不太明白如何将输入节点分解为多个由麦克劳林级数生成的具有不同非线性线连接的节点。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输入节点？如何轻松实现这一点？有什么想法吗？

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，我遇到了一堆编译错误和维度错误。我假设 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？

期望输出:

我期待用麦克劳林级数非线性展开来扩展CNN，如何让上面的实现正确高效？有什么可能的想法或方法吗？

Answer 1

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

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

该实现对来自卷积网络的形状为 (batch_size, input_dim=512) 的扁平张量的每个元素应用相同的泰勒展开。

UPDATE：正如我们在评论部分讨论的那样，这里有一些代码展示了如何修改您的函数 expandable_cnn 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工作（例如，您会获得良好的性能）。我只是根据我对你想要的解释提供了一个解决方案。

如何在keras中实现麦克劳林级数？

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