ValueError：形状（240000,28,28）和（2,512）未对齐：28（dim 2）！= 2（dim 0）
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【中文标题】ValueError：形状（240000,28,28）和（2,512）未对齐：28（dim 2）！= 2（dim 0）【英文标题】：ValueError: shapes (240000,28,28) and (2,512) not aligned: 28 (dim 2) != 2 (dim 0) 【发布时间】：2021-07-30 03:22:44 【问题描述】：我正在制作一个 CNN，但我遇到了矩阵未对齐的错误，我理解该错误，但我不知道如何修复它。代码如下：
import numpy as np
import nnfs
import emnist
import os
import cv2
import pickle
import copy
nnfs.init()
# Dense layer
class Layer_Dense:
# Layer initialization
def __init__(self, n_inputs, n_neurons,
weight_regularizer_l1=0, weight_regularizer_l2=0,
bias_regularizer_l1=0, bias_regularizer_l2=0):
# Initialize weights and biases
self.weights = 0.01 * np.random.randn(n_inputs, n_neurons)
self.biases = np.zeros((1, n_neurons))
# Set regularization strength
self.weight_regularizer_l1 = weight_regularizer_l1
self.weight_regularizer_l2 = weight_regularizer_l2
self.bias_regularizer_l1 = bias_regularizer_l1
self.bias_regularizer_l2 = bias_regularizer_l2
# Forward pass
def forward(self, inputs, training):
# Remember input values
self.inputs = inputs
# Calculate output values from inputs, weights and biases
self.output = np.dot(inputs, self.weights) + self.biases
# Backward pass
def backward(self, dvalues):
# Gradients on parameters
self.dweights = np.dot(self.inputs.T, dvalues)
self.dbiases = np.sum(dvalues, axis=0, keepdims=True)
# Gradients on regularization
# L1 on weights
if self.weight_regularizer_l1 > 0:
dL1 = np.ones_like(self.weights)
dL1[self.weights < 0] = -1
self.dweights += self.weight_regularizer_l1 * dL1
# L2 on weights
if self.weight_regularizer_l2 > 0:
self.dweights += 2 * self.weight_regularizer_l2 * \
self.weights
# L1 on biases
if self.bias_regularizer_l1 > 0:
dL1 = np.ones_like(self.biases)
dL1[self.biases < 0] = -1
self.dbiases += self.bias_regularizer_l1 * dL1
# L2 on biases
if self.bias_regularizer_l2 > 0:
self.dbiases += 2 * self.bias_regularizer_l2 * \
self.biases
# Gradient on values
self.dinputs = np.dot(dvalues, self.weights.T)
# Retrieve layer parameters
def get_parameters(self):
return self.weights, self.biases
# Set weights and biases in a layer instance
def set_parameters(self, weights, biases):
self.weights = weights
self.biases = biases
# Dropout
class Layer_Dropout:
# Init
def __init__(self, rate):
# Store rate, we invert it as for example for dropout
# of 0.1 we need success rate of 0.9
self.rate = 1 - rate
# Forward pass
def forward(self, inputs, training):
# Save input values
self.inputs = inputs
# If not in the training mode - return values
if not training:
self.output = inputs.copy()
return
# Generate and save scaled mask
self.binary_mask = np.random.binomial(1, self.rate,size=inputs.shape) / self.rate
# Apply mask to output values
self.output = inputs * self.binary_mask
# Backward pass
def backward(self, dvalues):
# Gradient on values
self.dinputs = dvalues * self.binary_mask
#Input "layer"
class Layer_Input:
# Forward pass
def forward(self, inputs, training):
self.output = inputs
# ReLU activation
class Activation_ReLU:
# Forward pass
def forward(self, inputs, training):
# Remember input values
self.inputs = inputs
# Calculate output values from inputs
self.output = np.maximum(0, inputs)
# Backward pass
def backward(self, dvalues):
# Since we need to modify original variable,
# let's make a copy of values first
self.dinputs = dvalues.copy()
# Zero gradient where input values were negative
self.dinputs[self.inputs <= 0] = 0
# Calculate predictions for outputs
def predictions(self, outputs):
return outputs
# Softmax activation
class Activation_Softmax:
# Forward pass
def forward(self, inputs, training):
# Remember input values
self.inputs = inputs
# Get unnormalized probabilities
exp_values = np.exp(inputs - np.max(inputs, axis=1,keepdims=True))
# Normalize them for each sample
probabilities = exp_values / np.sum(exp_values, axis=1,keepdims=True)
self.output = probabilities
# Backward pass
def backward(self, dvalues):
# Create uninitialized array
self.dinputs = np.empty_like(dvalues)
# Enumerate outputs and gradients
for index, (single_output, single_dvalues) in enumerate(zip(self.output, dvalues)):
# Flatten output array
single_output = single_output.reshape(-1, 1)
# Calculate Jacobian matrix of the output
jacobian_matrix = np.diagflat(single_output) - np.dot(single_output, single_output.T)
# Calculate sample-wise gradient
# and add it to the array of sample gradients
self.dinputs[index] = np.dot(jacobian_matrix,single_dvalues)
# Calculate predictions for outputs
def predictions(self, outputs):
return np.argmax(outputs, axis=1)
# Adam optimizer
class Optimizer_Adam:
# Initialize optimizer - set settings
def __init__(self, learning_rate=0.001, decay=0., epsilon=1e-7,
beta_1=0.9, beta_2=0.999):
self.learning_rate = learning_rate
self.current_learning_rate = learning_rate
self.decay = decay
self.iterations = 0
self.epsilon = epsilon
self.beta_1 = beta_1
self.beta_2 = beta_2
# Call once before any parameter updates
def pre_update_params(self):
if self.decay:
self.current_learning_rate = self.learning_rate * (1. / (1. + self.decay * self.iterations))
# Update parameters
def update_params(self, layer):
# If layer does not contain cache arrays,
# create them filled with zeros
if not hasattr(layer, 'weight_cache'):
layer.weight_momentums = np.zeros_like(layer.weights)
layer.weight_cache = np.zeros_like(layer.weights)
layer.bias_momentums = np.zeros_like(layer.biases)
layer.bias_cache = np.zeros_like(layer.biases)
# Update momentum with current gradients
layer.weight_momentums = self.beta_1 * layer.weight_momentums + (1 - self.beta_1) * layer.dweights
layer.bias_momentums = self.beta_1 * layer.bias_momentums + (1 - self.beta_1) * layer.dbiases
# Get corrected momentum
# self.iteration is 0 at first pass
# and we need to start with 1 here
weight_momentums_corrected = layer.weight_momentums / (1 - self.beta_1 ** (self.iterations + 1))
bias_momentums_corrected = layer.bias_momentums / (1 - self.beta_1 ** (self.iterations + 1))
# Update cache with squared current gradients
layer.weight_cache = self.beta_2 * layer.weight_cache + (1 - self.beta_2) * layer.dweights**2
layer.bias_cache = self.beta_2 * layer.bias_cache + (1 - self.beta_2) * layer.dbiases**2
# Get corrected cache
weight_cache_corrected = layer.weight_cache / (1 - self.beta_2 ** (self.iterations + 1))
bias_cache_corrected = layer.bias_cache / (1 - self.beta_2 ** (self.iterations + 1))
# Vanilla SGD parameter update + normalization
# with square rooted cache
layer.weights += -self.current_learning_rate * weight_momentums_corrected / (np.sqrt(weight_cache_corrected) + self.epsilon)
layer.biases += -self.current_learning_rate * bias_momentums_corrected / (np.sqrt(bias_cache_corrected) + self.epsilon)
# Call once after any parameter updates
def post_update_params(self):
self.iterations += 1
# Common loss class
class Loss:
# Regularization loss calculation
def regularization_loss(self):
# 0 by default
regularization_loss = 0
# Calculate regularization loss
# iterate all trainable layers
for layer in self.trainable_layers:
# L1 regularization - weights
# calculate only when factor greater than 0
if layer.weight_regularizer_l1 > 0:
regularization_loss += layer.weight_regularizer_l1 * np.sum(np.abs(layer.weights))
# L2 regularization - weights
if layer.weight_regularizer_l2 > 0:
regularization_loss += layer.weight_regularizer_l2 * np.sum(layer.weights * layer.weights)
# L1 regularization - biases
# calculate only when factor greater than 0
if layer.bias_regularizer_l1 > 0:
regularization_loss += layer.bias_regularizer_l1 * np.sum(np.abs(layer.biases))
# L2 regularization - biases
if layer.bias_regularizer_l2 > 0:
regularization_loss += layer.bias_regularizer_l2 * np.sum(layer.biases * layer.biases)
return regularization_loss
# Set/remember trainable layers
def remember_trainable_layers(self, trainable_layers):
self.trainable_layers = trainable_layers
# Calculates the data and regularization losses
# given model output and ground truth values
def calculate(self, output, y, *, include_regularization=False):
# Calculate sample losses
sample_losses = self.forward(output, y)
# Calculate mean loss
data_loss = np.mean(sample_losses)
# Add accumulated sum of losses and sample count
self.accumulated_sum += np.sum(sample_losses)
self.accumulated_count += len(sample_losses)
# If just data loss - return it
if not include_regularization:
return data_loss
# Return the data and regularization losses
return data_loss, self.regularization_loss()
# Calculates accumulated loss
def calculate_accumulated(self, *, include_regularization=False):
# Calculate mean loss
data_loss = self.accumulated_sum / self.accumulated_count
# If just data loss - return it
if not include_regularization:
return data_loss
# Return the data and regularization losses
return data_loss, self.regularization_loss()
# Reset variables for accumulated loss
def new_pass(self):
self.accumulated_sum = 0
self.accumulated_count = 0
# Cross-entropy loss
class Loss_CategoricalCrossentropy(Loss):
# Forward pass
def forward(self, y_pred, y_true):
# Number of samples in a batch
samples = len(y_pred)
# Clip data to prevent division by 0
# Clip both sides to not drag mean towards any value
y_pred_clipped = np.clip(y_pred, 1e-7, 1 - 1e-7)
# Probabilities for target values -
# only if categorical labels
if len(y_true.shape) == 1:
correct_confidences = y_pred_clipped[range(samples),y_true]
# Mask values - only for one-hot encoded labels
elif len(y_true.shape) == 2:
correct_confidences = np.sum(y_pred_clipped * y_true,axis=1)
# Losses
negative_log_likelihoods = -np.log(correct_confidences)
return negative_log_likelihoods
# Backward pass
def backward(self, dvalues, y_true):
# Number of samples
samples = len(dvalues)
# Number of labels in every sample
# We'll use the first sample to count them
labels = len(dvalues[0])
# If labels are sparse, turn them into one-hot vector
if len(y_true.shape) == 1:
y_true = np.eye(labels)[y_true]
# Calculate gradient
self.dinputs = -y_true / dvalues
# Normalize gradient
self.dinputs = self.dinputs / samples
# Softmax classifier - combined Softmax activation
# and cross-entropy loss for faster backward step
class Activation_Softmax_Loss_CategoricalCrossentropy():
# Backward pass
def backward(self, dvalues, y_true):
# Number of samples
samples = len(dvalues)
# If labels are one-hot encoded,
# turn them into discrete values
if len(y_true.shape) == 2:
y_true = np.argmax(y_true, axis=1)
# Copy so we can safely modify
self.dinputs = dvalues.copy()
# Calculate gradient
self.dinputs[range(samples), y_true] -= 1
# Normalize gradient
self.dinputs = self.dinputs / samples
# Common accuracy class
class Accuracy:
# Calculates an accuracy
# given predictions and ground truth values
def calculate(self, predictions, y):
# Get comparison results
comparisons = self.compare(predictions, y)
# Calculate an accuracy
accuracy = np.mean(comparisons)
# Add accumulated sum of matching values and sample count
self.accumulated_sum += np.sum(comparisons)
self.accumulated_count += len(comparisons)
# Return accuracy
return accuracy
# Calculates accumulated accuracy
def calculate_accumulated(self):
# Calculate an accuracy
accuracy = self.accumulated_sum / self.accumulated_count
# Return the data and regularization losses
return accuracy
# Reset variables for accumulated accuracy
def new_pass(self):
self.accumulated_sum = 0
self.accumulated_count = 0
# Accuracy calculation for classification model
class Accuracy_Categorical(Accuracy):
def __init__(self, *, binary=False):
# Binary mode?
self.binary = binary
# No initialization is needed
def init(self, y):
pass
# Compares predictions to the ground truth values
def compare(self, predictions, y):
if not self.binary and len(y.shape) == 2:
y = np.argmax(y, axis=1)
return predictions == y
# Model class
class Model:
def __init__(self):
# Create a list of network objects
self.layers = []
# Softmax classifier's output object
self.softmax_classifier_output = None
# Add objects to the model
def add(self, layer):
self.layers.append(layer)
#
# Set loss, optimizer and accuracy
def set(self, *, loss=None, optimizer=None, accuracy=None):
if loss is not None:
self.loss = loss
if optimizer is not None:
self.optimizer = optimizer
if accuracy is not None:
self.accuracy = accuracy
# Finalize the model
def finalize(self):
# Create and set the input layer
self.input_layer = Layer_Input()
# Count all the objects
layer_count = len(self.layers)
# Initialize a list containing trainable layers:
self.trainable_layers = []
# Iterate the objects
for i in range(layer_count):
# If it's the first layer,
# the previous layer object is the input layer
if i == 0:
self.layers[i].prev = self.input_layer
self.layers[i].next = self.layers[i+1]
# All layers except for the first and the last
elif i < layer_count - 1:
self.layers[i].prev = self.layers[i-1]
self.layers[i].next = self.layers[i+1]
# The last layer - the next object is the loss
# Also let's save aside the reference to the last object
# whose output is the model's output
else:
self.layers[i].prev = self.layers[i-1]
self.layers[i].next = self.loss
self.output_layer_activation = self.layers[i]
# If layer contains an attribute called "weights",
# it's a trainable layer -
# add it to the list of trainable layers
# We don't need to check for biases -
# checking for weights is enough
if hasattr(self.layers[i], 'weights'):
self.trainable_layers.append(self.layers[i])
# Update loss object with trainable layers
if self.loss is not None:
self.loss.remember_trainable_layers(self.trainable_layers)
# If output activation is Softmax and
# loss function is Categorical Cross-Entropy
# create an object of combined activation
# and loss function containing
# faster gradient calculation
if isinstance(self.layers[-1], Activation_Softmax) and isinstance(self.loss, Loss_CategoricalCrossentropy):
# Create an object of combined activation
# and loss functions
self.softmax_classifier_output = Activation_Softmax_Loss_CategoricalCrossentropy()
# Train the model
def train(self, X, y, *, epochs=1, batch_size=None,print_every=1, validation_data=None):
# Initialize accuracy object
self.accuracy.init(y)
# Default value if batch size is not being set
train_steps = 1
# Calculate number of steps
if batch_size is not None:
train_steps = len(X) // batch_size
# Dividing rounds down. If there are some remaining
# data but not a full batch, this won't include it
# Add `1` to include this not full batch
if train_steps * batch_size < len(X):
train_steps += 1
# Main training loop
for epoch in range(1, epochs+1):
# Print epoch number
print(f'epoch: epoch')
# Reset accumulated values in loss and accuracy objects
self.loss.new_pass()
self.accuracy.new_pass()
# Iterate over steps
for step in range(train_steps):
# If batch size is not set -
# train using one step and full dataset
if batch_size is None:
batch_X = X
batch_y = y
# Otherwise slice a batch
else:
batch_X = X[step*batch_size:(step+1)*batch_size]
atch_y = y[step*batch_size:(step+1)*batch_size]
# Perform the forward pass
output = self.forward(batch_X, training=True)
# Calculate loss
data_loss, regularization_loss = self.loss.calculate(output, batch_y,include_regularization=True)
loss = data_loss + regularization_loss
# Get predictions and calculate an accuracy
predictions = self.output_layer_activation.predictions(output)
accuracy = self.accuracy.calculate(predictions,batch_y)
# Perform backward pass
self.backward(output, batch_y)
# Optimize (update parameters)
self.optimizer.pre_update_params()
for layer in self.trainable_layers:
self.optimizer.update_params(layer)
self.optimizer.post_update_params()
# Print a summary
if not step % print_every or step == train_steps - 1:
print(f'step: step, ' +
f'acc: accuracy:.3f, ' +
f'loss: loss:.3f (' +
f'data_loss: data_loss:.3f, ' +
f'reg_loss: regularization_loss:.3f), ' +
f'lr: self.optimizer.current_learning_rate')
# Get and print epoch loss and accuracy
epoch_data_loss, epoch_regularization_loss = self.loss.calculate_accumulated(include_regularization=True)
epoch_loss = epoch_data_loss + epoch_regularization_loss
epoch_accuracy = self.accuracy.calculate_accumulated()
print(f'training, ' +
f'acc: epoch_accuracy:.3f, ' +
f'loss: epoch_loss:.3f (' +
f'data_loss: epoch_data_loss:.3f, ' +
f'reg_loss: epoch_regularization_loss:.3f), ' +
f'lr: self.optimizer.current_learning_rate')
# If there is the validation data
if validation_data is not None:
# Evaluate the model:
self.evaluate(*validation_data,batch_size=batch_size)
# Evaluates the model using passed-in dataset
def evaluate(self, X_val, y_val, *, batch_size=None):
# Default value if batch size is not being set
validation_steps = 1
# Calculate number of steps
if batch_size is not None:
validation_steps = len(X_val) // batch_size
# Dividing rounds down. If there are some remaining
# data but not a full batch, this won't include it
# Add `1` to include this not full batch
if validation_steps * batch_size < len(X_val):
validation_steps += 1
# Reset accumulated values in loss
# and accuracy objects
self.loss.new_pass()
self.accuracy.new_pass()
# Iterate over steps
for step in range(validation_steps):
# If batch size is not set -
# train using one step and full dataset
if batch_size is None:
batch_X = X_val
batch_y = y_val
# Otherwise slice a batch
else:
batch_X = X_val[step*batch_size:(step+1)*batch_size]
batch_y = y_val[step*batch_size:(step+1)*batch_size]
# Perform the forward pass
output = self.forward(batch_X, training=False)
# Calculate the loss
self.loss.calculate(output, batch_y)
# Get predictions and calculate an accuracy
predictions = self.output_layer_activation.predictions(output)
self.accuracy.calculate(predictions, batch_y)
# Get and print validation loss and accuracy
validation_loss = self.loss.calculate_accumulated()
validation_accuracy = self.accuracy.calculate_accumulated()
# Print a summary
print(f'validation, ' +
f'acc: validation_accuracy:.3f, ' +
f'loss: validation_loss:.3f')
# Predicts on the samples
def predict(self, X, *, batch_size=None):
# Default value if batch size is not being set
prediction_steps = 1
# Calculate number of steps
if batch_size is not None:
prediction_steps = len(X) // batch_size
# Dividing rounds down. If there are some remaining
# data but not a full batch, this won't include it
# Add `1` to include this not full batch
if prediction_steps * batch_size < len(X):
prediction_steps += 1
# Model outputs
output = []
# Iterate over steps
for step in range(prediction_steps):
# If batch size is not set -
# train using one step and full dataset
if batch_size is None:
batch_X = X
# Otherwise slice a batch
else:
batch_X = X[step*batch_size:(step+1)*batch_size]
# Perform the forward pass
batch_output = self.forward(batch_X, training=False)
# Append batch prediction to the list of predictions
output.append(batch_output)
# Stack and return results
return np.vstack(output)
# Performs forward pass
def forward(self, X, training):
# Call forward method on the input layer
# this will set the output property that
# the first layer in "prev" object is expecting
self.input_layer.forward(X, training)
# Call forward method of every object in a chain
# Pass output of the previous object as a parameter
for layer in self.layers:
layer.forward(layer.prev.output, training)
# "layer" is now the last object from the list,
# return its output
# Performs backward pass
def backward(self, output, y):
# If softmax classifier
if self.softmax_classifier_output is not None:
# First call backward method
# on the combined activation/loss
# this will set dinputs property
self.softmax_classifier_output.backward(output, y)
# Since we'll not call backward method of the last layer
# which is Softmax activation
# as we used combined activation/loss
# object, let's set dinputs in this object
self.layers[-1].dinputs = self.softmax_classifier_output.dinputs
# Call backward method going through
# all the objects but last
# in reversed order passing dinputs as a parameter
for layer in reversed(self.layers[:-1]):
layer.backward(layer.next.dinputs)
return
# First call backward method on the loss
# this will set dinputs property that the last
# layer will try to access shortly
self.loss.backward(output, y)
# Call backward method going through all the objects
# in reversed order passing dinputs as a parameter
for layer in reversed(self.layers):
layer.backward(layer.next.dinputs)
# Retrieves and returns parameters of trainable layers
def get_parameters(self):
# Create a list for parameters
parameters = []
# Iterable trainable layers and get their parameters
for layer in self.trainable_layers:
parameters.append(layer.get_parameters())
# Return a list
return parameters
#Updates the model with new parameters
def set_parameters(self, parameters):
# Iterate over the parameters and layers
# and update each layers with each set of the parameters
for parameter_set, layer in zip(parameters,self.trainable_layers):
layer.set_parameters(*parameter_set)
# Saves the parameters to a file
def save_parameters(self, path):
# Open a file in the binary-write mode
# and save parameters into it
with open(path, 'wb') as f:
pickle.dump(self.get_parameters(), f)
# Loads the weights and updates a model instance with them
def load_parameters(self, path):
# Open file in the binary-read mode,
# load weights and update trainable layers
with open(path, 'rb') as f:
self.set_parameters(pickle.load(f))
# Saves the model
def save(self, path):
# Make a deep copy of current model instance
model = copy.deepcopy(self)
# Reset accumulated values in loss and accuracy objects
model.loss.new_pass()
model.accuracy.new_pass()
# Remove data from the input layer
# and gradients from the loss object
model.input_layer.__dict__.pop('output', None)
model.loss.__dict__.pop('dinputs', None)
# For each layer remove inputs, output and dinputs properties
for layer in model.layers:
for property in ['inputs', 'output', 'dinputs','dweights', 'dbiases']:
layer.__dict__.pop(property, None)
# Open a file in the binary-write mode and save the model
with open(path, 'wb') as f:
pickle.dump(model, f)
# Loads and returns a model
@staticmethod
def load(path):
# Open file in the binary-read mode, load a model
with open(path, 'rb') as f:
model = pickle.load(f)
# Return a model
return model
# Create dataset
X, y = emnist.extract_training_samples('digits')
X_test, y_test = emnist.extract_test_samples('digits')
# Instantiate the model
model = Model()
# Add layers
model.add(Layer_Dense(2, 512, weight_regularizer_l2=5e-4,bias_regularizer_l2=5e-4))
model.add(Activation_ReLU())
model.add(Layer_Dropout(0.1))
model.add(Layer_Dense(512, 3))
model.add(Activation_Softmax())
# Set loss, optimizer and accuracy objects
model.set(
loss=Loss_CategoricalCrossentropy(),
optimizer=Optimizer_Adam(learning_rate=0.05, decay=5e-5),
accuracy=Accuracy_Categorical()
)
# Finalize the model
model.finalize()
# Train the model
model.train(X, y, validation_data=(X_test, y_test),epochs=10000, print_every=100)
这是我在崇高文本中遇到的错误：
epoch: 1
Traceback (most recent call last):
File "/media/luke/New Volume/final project/untitled.py", line 654, in <module>
model.train(X, y, validation_data=(X_test, y_test),epochs=10000, print_every=100)
File "/media/luke/New Volume/final project/untitled.py", line 430, in train
output = self.forward(batch_X, training=True)
File "/media/luke/New Volume/final project/untitled.py", line 545, in forward
layer.forward(layer.prev.output, training)
File "/media/luke/New Volume/final project/untitled.py", line 29, in forward
self.output = np.dot(inputs, self.weights) + self.biases
File "/home/luke/.local/lib/python3.8/site-packages/nnfs/core.py", line 22, in dot
return orig_dot(*[a.astype('float64') for a in args], **kwargs).astype('float32')
File "<__array_function__ internals>", line 5, in dot
ValueError: shapes (240000,28,28) and (2,512) not aligned: 28 (dim 2) != 2 (dim 0)
如您所见，当尝试执行 numpy 点积时，它到达 epoch 1，然后无法执行。
感谢您的帮助
谢谢:)
【问题讨论】：
【参考方案1】：首先，您应该展平输入，使其形状为(240000, 28*28) = (240000, 784)。之后，问题出在这一行：
model.add(Layer_Dense(2, 512, weight_regularizer_l2=5e-4,bias_regularizer_l2=5e-4))
您将输入大小设置为 2，此时它应该是 784，这是每个图像中的像素数（假设您使用的是 MNIST）。
model.add(Layer_Dense(784, 512, weight_regularizer_l2=5e-4,bias_regularizer_l2=5e-4))
如果您的输入被展平，应该可以正常工作。
编辑：为了使您的输入变平，我将使用np.reshape，如此处所示https://***.com/a/18758049/11777402。
X.reshape(240000, 784)
【讨论】：
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