CNTKx is a deep learning library that builds on and extends Microsoft Cognitive Toolkit CNTK. Despite the last planned release of cntk 2.7, cntkx will continue to be in active development, more models and pre-built components coming soon!
Feel free to open an issue for any request or a PR to contribute :)
cntkx is written in pure python and cntk is a dependency to it. Please get a working installation of cntk first. Then:
pip install cntkx
cntkx only works with python>=3.6
| ops | Description |
|---|---|
scalar |
cast tensor to scalar (1,) |
cumsum |
Cumulative summation along axis |
upsample |
Upsample by 2x (for image) |
centre_crop |
Crop centre of image |
swish |
Activation |
hardmax |
Activation |
erf |
Error function |
gelu |
Gaussian Error Linear Unit function |
gelu_fast |
fast approximation of Gaussian Error Linear Unit function |
sequence.pad |
Pad at start or end of sequence axis |
sequence.length |
length of sequence |
sequence.position |
position of every sequence element |
sequence.stride |
strides across sequential axis |
sequence.join |
joins two sequence along their sequential axis |
sequence.window |
creates non-overlapping window along the sequence axis |
sequence.reverse |
reverses the items along the dynamic sequence axis |
sequence.reduce_mean |
calculates the mean along the dynamic sequence axis |
random.sample |
Samples an unnormalised log probability distribution |
random.sample_top_k |
Samples from the top_k of an unnormalised log probability distribution |
batchmatmul |
Batch Matrix Multiplication on a static batch axis, similar to tf.matmul |
| Layers | Description |
|---|---|
QRNN |
Quasi-Recurrent Neural Network |
Recurrence |
With option to apply VariationalDroppout |
PyramidalBiRecurrence |
Pyramidal bi-directional recurrence |
VariationalDropout |
Single binary dropout mask for entire sequence |
SinusoidalPositionalEmbedding |
Non-learnable positional embedding (no max sequence length) |
PositionalEmbedding |
Learnable Positional Embedding (used in BERT) |
BertEmbeddings |
BERT Embeddings (word + token_type + positional) |
BertPooler |
Pooler used in BERT |
SpatialPyramidPooling |
Fixed pooled representation regardless of image input size |
GatedLinearUnit |
Gated Convolutional Neural Network |
ScaledDotProductAttention |
Attention used in BERT and Transformer (aka 'attention is all you need') |
MultiHeadAttention |
Attention used in BERT and Transformer (aka 'attention is all you need') |
GaussianWindowAttention |
Windowed attention instead of conventional attention where everything is attended at the same time |
SequentialMaxPooling |
Max pool across sequential axis and static axes |
SequentialAveragePooling |
Average pool across sequential axis and static axes |
vFSMN |
Vectorised Feedforward Sequential Memory Networks |
cFSMN |
Compact Feedforward Sequential Memory Networks |
BiRecurrence |
BiRecurrence recurrent layer with weight tying option to half parameter requirement |
| Blocks | Description |
|---|---|
WeightDroppedLSTM |
A form of regularised LSTM |
IndyLSTM |
A parameter efficient form of LSTM |
IndRNN |
a RNN with long memory and can be stacked deeply |
| Loss | Description |
|---|---|
gaussian_mdn_loss |
loss function when using Mixture density network |
focal_loss_with_softmax |
A kind of cross entropy that handles extreme class imbalance |
cross_entropy_with_softmax |
Added label smoothing regularisation in cross entropy with softmax |
| Models | Description |
|---|---|
VGG |
Image Classification |
UNET |
Semantic Segmentation |
Transformer |
Language Modelling |
MDN |
Mixture Density Networks |
| Pre-trained models | Description |
|---|---|
Bert |
Bidirectional Encoder Representations from Transformers |
| fwd_wt103.hdf5 | The weight parameters of the fastai's pytorch model. To be used to initialise PretrainedWikitext103LanguageModel |
| fwd_wt103.cntk | The converted cntk model of fastai's pytorch model. To be used with C.load_model |
| fwd_wt103.onnx | The converted ONNX model of fastai's pytorch model. |
| Learners | Description |
|---|---|
CyclicalLearningRate |
a method to eliminate the need to find best value and schedule for learning rate |
| Misc | Description |
|---|---|
CTCEncoder |
Helper class to convert data into a format acceptable for cntk's ctc implementation |
This library is implemented in pure cntk python API. For help in cntk c#, you can refer to the two repository deep-learning-with-csharp-and-cntk and DeepBelief_Course4_Examples.
For the F# wrapper of CNTK please visit FsCNTK, it also contains some example implementations like seq2seq, autoencoder, LSTM, GAN.
2019-09-06.
Add a wrapper to create a bidirectional recurrent layer using BiRecurrence. Included in the implementation
is an option to half the number of parameters required by bidirectional recurrent layer. This is done
by only using one recurrent unit to do both forward and backward computation instead of the usual two.
A forward and backward token is used to initialise the hidden state so that the recurrent unit can tell
the directionality.
More details can be found in the paper Efficient Bidirectional Neural Machine Translation
Example:
a = C.sequence.input_variable(10)
b = BiRecurrence(LSTM(100), weight_tie=True)(a)
assert b.shape == (200, )
2019-08-29.
CNTK implementation of Fast AI's Universal Language ModelFine-tuning (ULMFiT) English model has been added. This language model was trained on Wikitext-103 and can be used as a base model for any downstream language task.
It is also much more efficient to run compare to BERT and other Transformer language models.
For more details, you can refer to the original paper here
Example:
vocab_size = 238462
converted_hdf5_model_file_path = 'PATH/fwd_wt103.hdf5' # this is not the original pytorch model
lm = PretrainedWikitext103LanguageModel(converted_hdf5_model_file_path)
a = C.sequence.input_variable(vocab_size)
prediction = lm(a) # next-word-prediction
features = prediction.features # features of tokens
assert prediction.shape == (vocab_size, )
assert features.shape == (400, )
| Model | Description |
|---|---|
| fwd_wt103.hdf5 | The weight parameters of the fastai's pytorch model. To be used to initialise PretrainedWikitext103LanguageModel |
| fwd_wt103.cntk | The converted cntk model of fastai's pytorch model. To be used with C.load_model |
| fwd_wt103.onnx | The converted ONNX model of fastai's pytorch model. |
| itos_wt103.pkl | Tokens used in pretrained model |
2019-08-08.
Added two cntk implementation of gelu activation function. gelu activation is used in BERT
and OpenAI's GPT and GPT-2.
Gaussian Error Linear Unit (GELU), a high-performing neuralnetwork activation function. The GELU nonlinearity is the expected transforma-tion of a stochastic regularizer which randomly applies the identity or zero mapto a neuron’s input. The GELU nonlinearity weights inputs by their magnitude,rather than gates inputs by their sign as in ReLUs.
For more detail please refer to Gaussian Error Linear Units (GELU) by Hendrycks and Gimpel.
2019-07-04.
Calculates the mean along the dynamic sequential axis in CNTK.
2019-06-26.
Allows the sequence items along the sequence axis to be reversed. This is useful when you want to create a Bi-directional Auto-Regressive layer because using UnfoldFrom does not work with Recurrence(go_backwards=True) and will result in a ValueError.
Example:
import cntk as C
import cntkx as Cx
from cntk.layers import Recurrence, UnfoldFrom, LSTM
hidden_dim = 50
start_token = C.Constant(0, shape=(hidden_dim,))
a = C.sequence.input_variable(1, name='seq1')
b = UnfoldFrom(Recurrence(LSTM(hidden_dim), go_backwards=True))(start_token, a)
n = [np.random.random((10, hidden_dim)).astype(np.float32),]
# This raise 'ValueError: It is not allowed to have multiple different stepping directions in the same loop'
b.eval({b.arguments[0]: n})
The workaround would be:
import cntk as C
import cntkx as Cx
from cntk.layers import Recurrence, UnfoldFrom, LSTM
hidden_dim = 50
start_token = C.Constant(0, shape=(hidden_dim,))
a = C.sequence.input_variable(1, name='seq1')
a_reversed = Cx.sequence.reverse(a)
b = UnfoldFrom(Recurrence(LSTM(hidden_dim)))(start_token, a_reversed) # remove go_backwards=True
n = [np.random.random((10, hidden_dim)).astype(np.float32),]
b.eval({b.arguments[0]: n}) # this will run just fine
2019-06-21.
CNTK implementation of that allows sampling of the top_k unnormalised log-probabilities distribution. This is useful text (or sequence) generation where it is known that greedy decoding will cause text degeneration.
For more details on this, please refer to A curious case of neural text degeneration
Example:
import cntk as C
import cntkx as Cx
a = C.input_variable(5)
b = Cx.random.sample_top_k(a, k=3, num_classes=5)
n = np.array([[1, 2, 3, 4, 5],] * 1000)
results = b.eval({a: n})
assert np.sum(results[:, :2]) == 0
assert np.sum(results[:, 2:]) == 1000
2019-05-04.
CNTK implementation of Bidirectional vectorised Feedforward Sequential Memory Network (vFSMN) and Compact Feedforward Sequential Memory Network (cFSMN).
FSMN is a standard fully-connected feedforward neural network equipped with some learnable memory blocks in its hidden layers. The memory blocks use a tapped-delay line structure to encode the long context information into a fixed-size representation as short-term memory mechanism.
The authors claim that FSMNs can be learned much more reliably and faster than RNNs or LSTMs due to the inherent non-recurrent model structure while significantly outperforming RNNs in language and speech modeling.
For more details please refer to Feedforward Sequential Memory Networks: A New Structure to Learn Long-term Dependency
cFSMN is a compact version of FSMN that can result in a reduction of up to 60% in model size and speed up the learning by more than 7 times while still significantly outperforming the popular bi-direction LSTMs for both frame-level cross-entropy (CE) criterion based training and MMI based sequence training.
For more details please refer to "Compact Feedforward Sequential Memory Networks for Large VocabularyContinuous Speech Recognition" by Zhang, et al.
Example:
import cntk as C
from cntkx.layers import vFSMN, cFSMN
# unidirectional vFSMN (only past conext used)
a = C.sequence.input_variable(10)
b = vFSMN(100, C.relu, num_past_context=3, num_future_context=0)(a)
assert b.shape == (100,)
# bidirectional vFSMN (enable both past and future context)
a = C.sequence.input_variable(10)
b = vFSMN(120, C.relu, num_past_context=3, num_future_context=3)(a)
assert b.shape == (120,)
# bidirectional cFSMN (enable both past and future context)
a = C.sequence.input_variable(100)
b = cFSMN(120, 50, C.relu, num_past_context=3, num_future_context=3)(a)
assert b.shape == (120,)
2019-04-19.
CNTK's CTC implementation requires that data be formatted in a particular way that's typically in acoustic modeling but unusual in other applications. So class provides an easy way to convert data between what users typically expect and what cntk demands.
Example: labels = ['a', 'b', 'c'] encoder = CTCEncoder(labels)
labels_tensor = C.sequence.input_variable(len(encoder.classes_)) # number of classes = 4
input_tensor = C.sequence.input_variable(100)
labels_graph = cntk.labels_to_graph(labels_tensor)
network_out = model(input_tensor)
fb = C.forward_backward(labels_graph, network_out, blankTokenId=encoder.blankTokenId)
ground_truth = ['a', 'b', 'b', 'b', 'c']
seq_length = 10 # must be the same length as the sequence length in network_out
fb.eval({input_tensor: [...],
labels_tensor: [encoder.transform(ground_truth, seq_length=seq_length)]})
2019-04-14.
Added Label Smoothing Regularization in cross_entropy_with_softmax.
Added sequence.window that creates non-overlapping window along the sequence axis thereby reducing the
sequence length and increasing the dimension by the same factor.
Implemented a convenience layer used in acoustic modelling known as PyramidalBiRecurrence. Used to create
pyramidal bi-directional LSTM (BLSTM) found in "Listen, attend and spell" by Chan et al. (https://arxiv.org/abs/1508.01211)
Typically used to down sample the sequence length to make memory and runtime manageable.
2019-04-08.
Added a new op called join where two sequence tensors can be joined along with sequence axis forming a longer sequence.
2019-04-08.
Add average pooling layer that works with sequential axis. Current cntk AveragePooling doesn't pool across sequence elements.
Example on cntkx.layers.SequentialAveragePooling
# rgb image of height 25 and variable width
a = C.sequence.input_variable((3, 25))
# Convolute across image with (3, 3) kernel with stride (1, 1)
b = C.layers.SequentialConvolution(filter_shape=(3, 3), num_filters=16, stride=(1, 1), pad=True)(a)
assert b.shape == (16, 25)
# max pool (2,2) in height and width with stride (2,2) in height and width, no padding
c = SequentialAveragePooling(filter_shape=(2, 2), strides=(2, 2), pad=False)(b)
assert c.shape == (16, 12)
2019-04-07.
Cx.sequence.stride enables striding across the sequential axis, selecting every integer items along the sequence.
Cx.scalar converts tensor into scalar of shape (1,)
2019-04-06.
CNTK implementation of Independently Recurrent Long Short-term Memory cells: IndyLSTMs by Gonnet and Deselaers, and Independently Recurrent Neural Network (IndRNN): Building A Longer andDeeper RNN by Li, et al.
Both IndyLSTM and IndRNN have hidden-to-hidden weights that are diagonal matrix instead of the usual full matrix.
Thus neurons in each layer are independent from each other, and the cross-channel information is
obtained through stacking multiple layers.
IndRNN allows for the use of C.relu activation thus allowing multiple IndRNN layers to be stacked together deeply.
IndyLSTM has parameters linear to the number of nodes in the linear, as opposed to standard LSTM that is quadratic
making IndyLSTM potentially faster and smaller as a model.
Authors of both IndRNN and IndyLSTM have claimed performance as good as or even better than regular LSTMs.
Example:
import cntk as C
from cntkx.layers import IndyLSTM, IndRNN, Recurrence
a = C.sequence.input_variable(10)
b = Recurrence(IndRNN(20))(a)
c = Recurrence(IndyLSTM(20))(a)
assert b.shape == c.shape == (20,)
2019-03-24.
Add max pooling layer that works with sequential axis. Current cntk MaxPooling doesn't pool across sequence elements.
Example on cntkx.layers.SequentialMaxPooling
# rgb image of height 25 and variable width
a = C.sequence.input_variable((3, 25))
# Convolute across image with (3, 3) kernel with stride (1, 1)
b = C.layers.SequentialConvolution(filter_shape=(3, 3), num_filters=16, stride=(1, 1), pad=True)(a)
assert b.shape == (16, 25)
# max pool (2,2) in height and width with stride (2,2) in height and width, no padding
c = SequentialMaxPooling(filter_shape=(2, 2), strides=(2, 2), pad=False)(b)
assert c.shape == (16, 12)
2019-03-18.
Cyclical learning rate (CLR) is an implementation to that practically eliminates the need to experimentally find the best values and schedule for the global learning rates.
Instead of monotonically decreasing the learning rate, this method lets the learning
rate cyclically vary between reasonable boundary values. Training with
cyclical learning rates instead of fixed values achieves improved classification
accuracy without a need to tune and often in fewer iterations.
More details can be found in Cyclical Learning Rates for Training Neural Networks by Leslie N. Smith
This CLR implementation can be used with the cntk training loop by adding only two lines of code:
model = C.layers.Dense(10)(C.input_variable(10))
sgd_momentum = C.momentum_sgd(model.parameters, 0.1, 0.9)
clr = CyclicalLeaningRate(sgd_momentum, minibatch_size=32) # first line of code
for epoch in range(10):
for batch in range(100):
trainer.train_minibatch(...)
clr.batch_step() # second line of code (to be called after every training update)
2019-03-12.
Added Batch Matrix Multiplication. This implementation is similar to tensorflow.matmul.
Example:
a = C.sequence.input_variable((3, 4, 5)) # batch matrix
b = C.sequence.input_variable((3, 5, 6)) # batch matrix
c = Cx.batchmatmul(a, b)
assert c.shape == (3, 4, 6) # 3 is treated as a batch axis
2019-03-10.
BERT, the state-of-the-art language model is now available as a CNTK pretrained model.
Currently, it is only tested to work with BERT-Base, Uncased (uncased_L-12_H-768_A-12) and can be
downloaded from Google AI
When you have downloaded BERT-Base, Uncased, there should be 5 files inside. You will need to .zip
three of those files into a tensorflow checkpoint file before you can load it into cntkx.
Those three files are: bert_model.ckpt.data-00000-of-00001, bert_model.ckpt.index, bert_model.ckpt.meta.
Then rename the extension of .zip into .ckpt and you are good to go.
Example below
text_tensor = C.sequence.input_variable(30522)
token_type_tensor = C.sequence.input_variable(2)
filepath_to_tf_bert_model = "YOUR_FILE_DIRECTORY/bert_model.ckpt"
model = Cx.layers.PreTrainedBertModel(filepath_to_tf_bert_model, num_heads=12, dropout_rate=0.1)
b = model(text_tensor, token_type_tensor)
assert b.shape == (768,)
For more details about BERT, you can find the original paper here, and some useful resources here and here.
Note: It goes without saying also that to use these pre-trained models you will need to have tensorflow installed since we are convert them from tensorflow models.
2019-03-06.
CNTK implementation of PositionalEmbedding, BertEmbeddings and tf-to-cntk PreTrainedBertEmbeddings.
BERT is a state-of-the-art language model from Google AI, more details can be found in
BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding.
Google AI's pre-trained BERT tensorflow model can be downloaded here.
Tensorflow would need to be installed in your environment if you intend to use PreTrainedBertEmbeddings, which
takes a tensorflow model and convert it cntk.
Example for PositionalEmbedding
a = C.sequence.input_variable(12)
b = PositionalEmbedding(max_seq_length, hidden_dim)(a)
assert b.shape == (hidden_dim, )
Example for BertEmbeddings
text_tensor = C.sequence.input_variable(100)
token_type_tensor = C.sequence.input_variable(2)
b = BertEmbeddings(max_seq_length, hidden_dim, 0.1)(text_tensor, token_type_tensor)
assert b.shape == (hidden_dim, )
Example for PreTrainedBertEmbeddings
text_tensor = C.sequence.input_variable(30522)
token_type_tensor = C.sequence.input_variable(2)
filepath_to_tf_bert_model = "YOURFILEPATH"
embeddings = PreTrainedBertEmbeddings(filepath_to_tf_bert_model, 0.1, False)
b = embeddings(text_tensor, token_type_tensor)
assert b.shape == (768, )
2019-03-02.
CNTK implementation of Variational Dropout found in
A Theoretically Grounded Application of Dropout in Recurrent Neural Networks
and Weight Dropped LSTM proposed in a salesforce research paper
Regularizing and Optimizing LSTM Language Models.
Weight Dropped LSTM is a regularised LSTM that uses DropConnect on hidden-to-hidden weights as a form of recurrent
regularisation. It also include application of variational dropout on the inputs and outputs of the recurrent units
for further regularisation.
Variational Drpoout is a regularisation that uses same dropout mask at each time step
(i.e. across the dynamic sequence axis) as opposed to the naive application of C.layers.Dropout to a sequence
which will result in a different dropout mask for every tensor along the sequence axis.
import cntkx as Cx
from cntkx.layers import Recurrence, WeightDroppedLSTM
import cntk as C
dropconnect_rate = 0.2
variationaldrop_rate = 0.1
seq = C.sequence.input_variable(56)
b = Recurrence(WeightDroppedLSTM(20, dropconnect_rate),
variational_dropout_rate_input=variationaldrop_rate,
variational_dropout_rate_output=variationaldrop_rate)(seq)
assert b.shape == (100, )
seq_dropped = VariationalDropout(0.1)(seq)
assert seq_dropped.shape == seq.shape
2019-02-02.
CNTK implementation of Gated Linear Unit (Gated CNN) founnd in Facebook AI Research Lab's paper: Language Modeling with Gated Convolutional Networks. This paper applies a convolutional approach to language modelling with a novel Gated-CNN model.
import cntkx as Cx
import cntk as C
seq = C.sequence.input_variable(56)
hidden = Cx.layers.GatedLinearUnit(window=2, hidden_dim=100)(seq)
assert hidden.shape == (100, )
2019-01-21.
CNTK implementation of Focal Loss enables the training of highly accurate dense object detectors in the
presence of vast numbers of easy background examples or dataset with extreme class imbalance (e.g. 1:1000).
Focal Loss focuses training on a sparse set of hard examples and prevents the vast number of easy negatives from
overwhelm-ing the model during training.
For more details please refer to Focal Loss for Dense Object Detection
import cntkx as Cx
Cx.focal_loss_with_softmax([[0, 0, 0.8, 0.2]], [[0, 0, 1, 0]]).eval()
array([[0.31306446]], dtype=float32)
2019-01-18.
Gaussian window attention model was first introduced by Alex Graves in "Generating sequences with recurrent neural networks".
It uses a mixture of gaussian windows to attend to portions of the sequence as oppose to the widely used attention model introduced in "Neural machine translation by jointly learning to align and translate" by Bahdanau, et al. that attends to the entire sequence all at once.
Gaussian window attention is also directional in its attention on the context sequence. When modeling strongly ordered sequences, gaussian window attention will be a natural choice due to this inductive bias.
import cntk as C
import cntkx as Cx
seq1 = C.Axis.new_unique_dynamic_axis('seq1')
seq2 = C.Axis.new_unique_dynamic_axis('seq2')
encoded = C.sequence.input_variable(30, sequence_axis=seq1)
query = C.sequence.input_variable(28, sequence_axis=seq2)
a = Cx.layers.GaussianWindowAttention(10)(encoded, query)
assert a.shape == (30, )
"Generating sequences with recurrent neural networks" can be found here. "Neural machine translation by jointly learning to align and translate" can be found here.
2019-01-16.
Spatial pyramid pooling layer is a pooling layer than returns a fixed length representation regardless of the image size/scale. It is frequently used for multi-size image training. It reported SOTA classification results using a single full-image representation without fine-tuning. For more details on the paper "Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition" by K. He, X. Zhang, S. Ren, J. Sun, link here.
import cntk as C
import cntkx as Cx
n = np.random.random((3, 3, 32, 32)).astype(np.float32)
a = C.input_variable((3, 32, 32))
b = Cx.layers.SpatialPyramidPooling((1, 2, 4))(a)
assert b.shape == (3 * (4 * 4 + 2 * 2 + 1),) # representation not dependent on image size
2019-01-15.
Added sinusoidal positional embedding used in Transformer. For an accessible explanation of transformer, you may look up here.
import cntk as C
import cntkx as Cx
a = C.sequence.input_variable(10)
b = SinusoidalPositionalEmbedding()(a)
assert b.shape == (10, )
Added cntkx.ops.erf error function.
2019-01-12.
VGG is for image classification and UNET is for semantic segmentation. VGG is implemented for completeness sake and should not be used for any serious classification task.
Paper on VGG can be found here titled "Very Deep Convolutional Networks for Large-Scale Image Recognition"
Paper for UNET can be found here titled "U-Net: Convolutional Networks for Biomedical Image Segmentation"
VGG example:
import cntk as C
import cntkx as Cx
a = C.input_variable((3, 64, 64))
b = Cx.layers.VGG19(100)(a)
assert b.shape == (100,)
UNET example:
import cntk as C
import cntkx as Cx
a = C.input_variable((3, 512, 512))
b = Cx.layers.UNET(num_classes=10, base_num_filters=64, pad=True)(a)
assert b.shape == (10, 512, 512)
Convenience functions such as cntkx.ops.upsample and centre_crop have also been added.
cntkx.ops.upsample upsamples an image twice on each spatial dim. centre_crop crops a smaller image from
a bigger one in the centre given a reference smaller image.
The Transformer was first introduced in the paper 'Attention is all you need'. The architecture is based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. More recently, BERT which broke almost all SOTA language task is also based on transformer and self-attention.
import cntk as C
import cntkx as Cx
a = C.sequence.input_variable(512)
b = C.sequence.input_variable(512)
transformer = Cx.layers.Transformer() # using default settings
decoded = transformer(a, b)
assert decoded.shape == (512, )
2018-12-08.
The QRNN provides similar accuracy to the LSTM but can be betwen 2 and 17 times faster than the highly optimized NVIDIA cuDNN LSTM implementation depending on the use case.
More details please refer to the original paper here.
import cntk as C
import cntkx as Cx
input_seq = C.sequence.input_variable(input_dim)
prediction_seq = Cx.layers.QRNN(hidden_dim=50)(input_seq)
2018-12-07.
Added two new sequence ops. cntkx.ops.sequence.pad allows padding on the sequence axis and
cntkx.ops.sequence.length calculates the length of the sequence.
2018-12-05.
Mixture density networks are neural networks that can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. MDN are very useful when you need to map an input to several correct targets (aka. one-to-many problem).
Updated with Gaussian Mixture Density Network ops and loss function. Ops will allow you to extract mdn coefficients and sample from the network.
More details on mdn can be found in this paper by Christopher Bishop.
import cntk as C
import cntkx as Cx
input_tensor = C.input_variable(1, name="input_tensor")
target_tensor = C.input_variable(1, name="target_tensor")
# model
inner = Dense(50, activation=C.relu)(input_tensor)
inner = Dense(50, activation=C.relu)(inner)
prediction_tensor = Dense((ndim + 2) * nmix, activation=None)(inner)
sampled = Cx.sample_gaussian_mdn(prediction_tensor, nmix, ndim) # sampling node
loss = Cx.gaussian_mdn_loss(prediction_tensor, target_tensor, nmix=nmix, ndim=ndim) # loss function