I have tensor t with shape (Batch_Size x Dims) and another tensor v with shape (Vocab_Size x Dims). I'd like to produce a tensor d with shape (Batch_Size x Vocab_Size), such that d[i,j] = norm(t[i] - v[j]).
Doing this for a single tensor (no batches) is trivial: d = torch.norm(v - t), since t would be broadcast. How can I do this when the tensors have batches?
Insert unitary dimensions into v and t to make them (1 x Vocab_Size x Dims) and (Batch_Size x 1 x Dims) respectively. Next, take the broadcasted difference to get a tensor of shape (Batch_Size x Vocab_Size x Dims). Pass that to torch.norm along with the optional dim=2 argument so that the norm is taken along the last dimension. This will result in the desired (Batch_Size x Vocab_Size) tensor of norms.
d = torch.norm(v.unsqueeze(0) - t.unsqueeze(1), dim=2)
Edit: As pointed out by #KonstantinosKokos in the comments, due to the broadcasting rules used by numpy and pytorch, the leading unitary dimension on v does not need to be explicit. I.e. you can use
d = torch.norm(v - t.unsqueeze(1), dim=2)
I need to perform a multiplication in order to get a regressed value from my features i, x, y, z, n, like so:
# use only one sample
feats = np.array([i, x, y, z, n]).reshape(1, -1)
# scale
scaler = StandardScaler()
# fit
feat_scaled = scaler.fit_transform(feats)
# multiply and get regressed value --> 'reg' is regression object
val = feat_scaled # reg.coef_ + reg.intercept_
Where:
print (reg.coef_.shape, feat_scaled.shape)
(55,) (1, 5)
But i'm getting the error:
ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 55 is different from 5)
How do i fix this?
reg.coef_ shape is (5,) and therefore the dot product with feat_scaled cannot be perform. this array need to be reshape to (5, 1).
As follows:
feat_scaled # reg.coef_.reshape(-1,1)+ reg.intercept_
I've seen a few implementations of Intersection over Union in Keras Tensorflow, but they all use inputs that represent the contents of the box regions. The network I'm working with passes the four corners of the boxes themselves. How can I implement the algorithm this way with Keras backend?
My attempt:
def IoU():
# y_true: Tensor from the generator of shape (B, N, 5). The last value for each box is the state of the anchor (ignore, negative, positive).
# y_pred: Tensor from the network of shape (B, N, 4).
# box coordinates (in axis 2): [x1, y1, x2, y2]
def _IoU(y_true, y_pred):
anchor_state = y_true[:,:,-1]
regression = y_pred
regression_target = y_true[:,:,:-1]
indices = backend.where(keras.backend.equal(anchor_state, 1))
regression = backend.gather_nd(regression, indices)
regression_target = backend.gather_nd(regression_target, indices)
x1_pred = regression[:,0] # <- fails here
y1_pred = regression[:,1]
x2_pred = regression[:,2]
y2_pred = regression[:,3]
x1_true = regression_target[:,0]
y1_true = regression_target[:,1]
x2_true = regression_target[:,2]
y2_true = regression_target[:,3]
xA = keras.backend.maximum(x1_pred, keras.backend.transpose(x1_true))
yA = keras.backend.maximum(y1_pred, keras.backend.transpose(y1_true))
xB = keras.backend.maximum(x2_pred, keras.backend.transpose(x2_true))
yB = keras.backend.maximum(y2_pred, keras.backend.transpose(y2_true))
interArea = keras.backend.maximum((xB-xA+1),0)*keras.backend.maximum((yB-yA+1),0)
pred_area = (x2_pred-x1_pred+1)*(y2_pred-y1_pred+1)
truth_area = (x2_true-x1_true+1)*(y2_true-y1_true+1)
iou_arr = interArea/(pred_area + keras.backend.transpose(truth_area) - interArea)
iou = keras.backend.mean(iou_arr)
return iou
return _IoU
After some debugging I found that backend.gather_nd reduced regression and regression_target from shapes (?, ?, 4) to shapes (?, 4), hence the indexing for x1_pred.
However, when I try to use this as a metric for a compiled model, I get the following error during runtime:
ValueError: slice index 2 of dimension 1 out of bounds. for 'metrics/_IoU_1/strided_slice_4' (op: 'StridedSlice') with input shapes: [?,2], [2], [2], [2] and with computed input tensors: input[1] = <0 2>, input[2] = <0 3>, input[3] = <1 1>.
If anybody knows a way that I can implement this in Keras, or sees where I've gone wrong, I'd be forever grateful.
EDIT: I figured out that this was because the method was being called twice, once where (y_true, y_pred) were the box prediction/truth (and the last dimension was of shape 4), and then again where (y_true, y_pred) were the class prediction/truth (and I have only 2 classes).
I did get some negative values and some values > 1 for the returned iou values, so maybe I'm still doing something incorrectly?
I am trying to use Keras Dot and have the following errors.
Could you explain what I am doing wrong?
x1 = Input(shape=(2,4))
x2 = Input(shape=(4,))
y1 = dot([x1,x2], axes = (2,1))
modelA = Model(inputs=[x1, x2], outputs=y1)
a1 = np.arange(16).reshape(2,2,4)
a2 = np.array( [1,2,3,4] )
modelA.predict([a1,a2])
---->
ValueError: Error when checking : expected input_40 to have shape (None, 4) but
got array with shape (4, 1)
I am new to Keras, too. And the following is what I figured out after playing around with the Dot operation.
Firstly, the shape parameter of Input layer is NOT including the batch size. In your code, x2 = Input(shape=(4,)), so x2 is expecting the input data to be (None, 4), (None refers to batch size), but a2 is np.array([1,2,3,4]), the shape is (1, 4), hence the error message.
To get rid of the error you need to add the batch_size dimension to a2.
But then there is another problem, according to the doc of Dot, I think x1 and x2 should have the same batch size:
if applied to a list of two tensors a and b of shape (batch_size, n), the output will be a tensor of shape (batch_size, 1) where each entry i will be the dot product between a[i] and b[i].
So I manually match the batch size of a1 and a2, and the batch size of a1 is 2, so a2 needs to be np.array([[1,2,3,4],[1,2,3,4]])
Now you can have your desire result:
[[ 20. 60.]
[100. 140.]]
A few more words for beginners like me, the shape of x1 is (batch_size, 2, 4), the shape of x2 is (batch_size, 4), it seems that they are not compatible. Now is when the 'axes' parameter comes into play. In OP's code, axes=(2,1) means to dot x1's 3rd axis (0-indexed, it's the axis with the length of 4), with x2's 2nd axis(also the length is 4). So it will be [0,1,2,3]dot[1,2,3,4]=20, [4,5,6,7]dot[1,2,3,4]=60 ...
So, I'm using Michael Nielson's machine learning book as a reference for my code (it is basically identical): http://neuralnetworksanddeeplearning.com/chap1.html
The code in question:
def backpropagate(self, image, image_value) :
# declare two new numpy arrays for the updated weights & biases
new_biases = [np.zeros(bias.shape) for bias in self.biases]
new_weights = [np.zeros(weight_matrix.shape) for weight_matrix in self.weights]
# -------- feed forward --------
# store all the activations in a list
activations = [image]
# declare empty list that will contain all the z vectors
zs = []
for bias, weight in zip(self.biases, self.weights) :
print(bias.shape)
print(weight.shape)
print(image.shape)
z = np.dot(weight, image) + bias
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
# -------- backward pass --------
# transpose() returns the numpy array with the rows as columns and columns as rows
delta = self.cost_derivative(activations[-1], image_value) * sigmoid_prime(zs[-1])
new_biases[-1] = delta
new_weights[-1] = np.dot(delta, activations[-2].transpose())
# l = 1 means the last layer of neurons, l = 2 is the second-last, etc.
# this takes advantage of Python's ability to use negative indices in lists
for l in range(2, self.num_layers) :
z = zs[-1]
sp = sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
new_biases[-l] = delta
new_weights[-l] = np.dot(delta, activations[-l-1].transpose())
return (new_biases, new_weights)
My algorithm can only get to the first round backpropagation before this error occurs:
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 97, in stochastic_gradient_descent
self.update_mini_batch(mini_batch, learning_rate)
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 117, in update_mini_batch
delta_biases, delta_weights = self.backpropagate(image, image_value)
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 160, in backpropagate
z = np.dot(weight, activation) + bias
ValueError: shapes (30,50000) and (784,1) not aligned: 50000 (dim 1) != 784 (dim 0)
I get why it's an error. The number of columns in weights doesn't match the number of rows in the pixel image, so I can't do matrix multiplication. Here's where I'm confused -- there are 30 neurons used in the backpropagation, each with 50,000 images being evaluated. My understanding is that each of the 50,000 should have 784 weights attached, one for each pixel. But when I modify the code accordingly:
count = 0
for bias, weight in zip(self.biases, self.weights) :
print(bias.shape)
print(weight[count].shape)
print(image.shape)
z = np.dot(weight[count], image) + bias
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
count += 1
I still get a similar error:
ValueError: shapes (50000,) and (784,1) not aligned: 50000 (dim 0) != 784 (dim 0)
I'm just really confuzzled by all the linear algebra involved and I think I'm just missing something about the structure of the weight matrix. Any help at all would be greatly appreciated.
It looks like the issue is in your changes to the original code.
I’be downloaded example from the link you provided and it works without any errors:
Here is full source code I used:
import cPickle
import gzip
import numpy as np
import random
def load_data():
"""Return the MNIST data as a tuple containing the training data,
the validation data, and the test data.
The ``training_data`` is returned as a tuple with two entries.
The first entry contains the actual training images. This is a
numpy ndarray with 50,000 entries. Each entry is, in turn, a
numpy ndarray with 784 values, representing the 28 * 28 = 784
pixels in a single MNIST image.
The second entry in the ``training_data`` tuple is a numpy ndarray
containing 50,000 entries. Those entries are just the digit
values (0...9) for the corresponding images contained in the first
entry of the tuple.
The ``validation_data`` and ``test_data`` are similar, except
each contains only 10,000 images.
This is a nice data format, but for use in neural networks it's
helpful to modify the format of the ``training_data`` a little.
That's done in the wrapper function ``load_data_wrapper()``, see
below.
"""
f = gzip.open('../data/mnist.pkl.gz', 'rb')
training_data, validation_data, test_data = cPickle.load(f)
f.close()
return (training_data, validation_data, test_data)
def load_data_wrapper():
"""Return a tuple containing ``(training_data, validation_data,
test_data)``. Based on ``load_data``, but the format is more
convenient for use in our implementation of neural networks.
In particular, ``training_data`` is a list containing 50,000
2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
containing the input image. ``y`` is a 10-dimensional
numpy.ndarray representing the unit vector corresponding to the
correct digit for ``x``.
``validation_data`` and ``test_data`` are lists containing 10,000
2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
numpy.ndarry containing the input image, and ``y`` is the
corresponding classification, i.e., the digit values (integers)
corresponding to ``x``.
Obviously, this means we're using slightly different formats for
the training data and the validation / test data. These formats
turn out to be the most convenient for use in our neural network
code."""
tr_d, va_d, te_d = load_data()
training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
training_results = [vectorized_result(y) for y in tr_d[1]]
training_data = zip(training_inputs, training_results)
validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
validation_data = zip(validation_inputs, va_d[1])
test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
test_data = zip(test_inputs, te_d[1])
return (training_data, validation_data, test_data)
def vectorized_result(j):
"""Return a 10-dimensional unit vector with a 1.0 in the jth
position and zeroes elsewhere. This is used to convert a digit
(0...9) into a corresponding desired output from the neural
network."""
e = np.zeros((10, 1))
e[j] = 1.0
return e
class Network(object):
def __init__(self, sizes):
"""The list ``sizes`` contains the number of neurons in the
respective layers of the network. For example, if the list
was [2, 3, 1] then it would be a three-layer network, with the
first layer containing 2 neurons, the second layer 3 neurons,
and the third layer 1 neuron. The biases and weights for the
network are initialized randomly, using a Gaussian
distribution with mean 0, and variance 1. Note that the first
layer is assumed to be an input layer, and by convention we
won't set any biases for those neurons, since biases are only
ever used in computing the outputs from later layers."""
self.num_layers = len(sizes)
self.sizes = sizes
self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
self.weights = [np.random.randn(y, x)
for x, y in zip(sizes[:-1], sizes[1:])]
def feedforward(self, a):
"""Return the output of the network if ``a`` is input."""
for b, w in zip(self.biases, self.weights):
a = sigmoid(np.dot(w, a)+b)
return a
def SGD(self, training_data, epochs, mini_batch_size, eta,
test_data=None):
"""Train the neural network using mini-batch stochastic
gradient descent. The ``training_data`` is a list of tuples
``(x, y)`` representing the training inputs and the desired
outputs. The other non-optional parameters are
self-explanatory. If ``test_data`` is provided then the
network will be evaluated against the test data after each
epoch, and partial progress printed out. This is useful for
tracking progress, but slows things down substantially."""
if test_data: n_test = len(test_data)
n = len(training_data)
for j in xrange(epochs):
random.shuffle(training_data)
mini_batches = [
training_data[k:k+mini_batch_size]
for k in xrange(0, n, mini_batch_size)]
for mini_batch in mini_batches:
self.update_mini_batch(mini_batch, eta)
if test_data:
print "Epoch {0}: {1} / {2}".format(
j, self.evaluate(test_data), n_test)
else:
print "Epoch {0} complete".format(j)
def update_mini_batch(self, mini_batch, eta):
"""Update the network's weights and biases by applying
gradient descent using backpropagation to a single mini batch.
The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
is the learning rate."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
for x, y in mini_batch:
delta_nabla_b, delta_nabla_w = self.backprop(x, y)
nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
self.weights = [w-(eta/len(mini_batch))*nw
for w, nw in zip(self.weights, nabla_w)]
self.biases = [b-(eta/len(mini_batch))*nb
for b, nb in zip(self.biases, nabla_b)]
def backprop(self, x, y):
"""Return a tuple ``(nabla_b, nabla_w)`` representing the
gradient for the cost function C_x. ``nabla_b`` and
``nabla_w`` are layer-by-layer lists of numpy arrays, similar
to ``self.biases`` and ``self.weights``."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
# feedforward
activation = x
activations = [x] # list to store all the activations, layer by layer
zs = [] # list to store all the z vectors, layer by layer
for b, w in zip(self.biases, self.weights):
z = np.dot(w, activation)+b
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
# backward pass
delta = self.cost_derivative(activations[-1], y) * \
sigmoid_prime(zs[-1])
nabla_b[-1] = delta
nabla_w[-1] = np.dot(delta, activations[-2].transpose())
# Note that the variable l in the loop below is used a little
# differently to the notation in Chapter 2 of the book. Here,
# l = 1 means the last layer of neurons, l = 2 is the
# second-last layer, and so on. It's a renumbering of the
# scheme in the book, used here to take advantage of the fact
# that Python can use negative indices in lists.
for l in xrange(2, self.num_layers):
z = zs[-l]
sp = sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
nabla_b[-l] = delta
nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
return (nabla_b, nabla_w)
def evaluate(self, test_data):
"""Return the number of test inputs for which the neural
network outputs the correct result. Note that the neural
network's output is assumed to be the index of whichever
neuron in the final layer has the highest activation."""
test_results = [(np.argmax(self.feedforward(x)), y)
for (x, y) in test_data]
return sum(int(x == y) for (x, y) in test_results)
def cost_derivative(self, output_activations, y):
"""Return the vector of partial derivatives \partial C_x /
\partial a for the output activations."""
return (output_activations-y)
#### Miscellaneous functions
def sigmoid(z):
"""The sigmoid function."""
return 1.0/(1.0+np.exp(-z))
def sigmoid_prime(z):
"""Derivative of the sigmoid function."""
return sigmoid(z)*(1-sigmoid(z))
training_data, validation_data, test_data = load_data_wrapper()
net = Network([784, 30, 10])
net.SGD(training_data, 30, 10, 3.0, test_data=test_data)
Additional info:
However, I would recommend using one of existing frameworks, for example - Keras to don't reinvent the wheel
Also, it was checked with python 3.6:
Kudos on digging into Nielsen's code. It's a great resource to develop thorough understanding of NN principles. Too many people leap ahead to Keras without knowing what goes on under the hood.
Each training example doesn't get its own weights. Each of the 784 features does. If each example got its own weights then each weight set would overfit to its corresponding training example. Also, if you later used your trained network to run inference on a single test example, what would it do with 50,000 sets of weights when presented with just one handwritten digit? Instead, each of the 30 neurons in your hidden layer learns a set of 784 weights, one for each pixel, that offers high predictive accuracy when generalized to any handwritten digit.
Import network.py and instantiate a Network class like this without modifying any code:
net = network.Network([784, 30, 10])
..which gives you a network with 784 input neurons, 30 hidden neurons and 10 output neurons. Your weight matrices will have dimensions [30, 784] and [10, 30], respectively. When you feed the network an input array of dimensions [784, 1] the matrix multiplication that gave you an error is valid because dim 1 of the weight matrix equals dim 0 of the input array (both 784).
Your problem is not implementation of backprop but rather setting up a network architecture appropriate for the shape of your input data. If memory serves Nielsen leaves backprop as a black box in chapter 1 and doesn't dive into it until chapter 2. Keep at it, and good luck!