If you read the Keras documentation entry for Dense, you will see that this call:

Dense(16, input_shape=(5,3))

would result in a Dense network with 3 inputs and 16 outputs which would be applied independently for each of 5 steps. So, if D(x) transforms 3 dimensional vector to 16-d vector, what you'll get as output from your layer would be a sequence of vectors: [D(x[0,:]), D(x[1,:]),..., D(x[4,:])] with shape (5, 16). In order to have the behavior you specify you may first Flatten your input to a 15-d vector and then apply Dense:

model = Sequential()
model.add(Flatten(input_shape=(3, 2)))
model.add(Dense(16))
model.add(Activation('relu'))
model.add(Dense(4))
model.compile(loss='mean_squared_error', optimizer='SGD')

EDIT: As some people struggled to understand - here you have an explaining image:

Units:

The amount of "neurons", or "cells", or whatever the layer has inside it.

It's a property of each layer, and yes, it's related to the output shape (as we will see later). In your picture, except for the input layer, which is conceptually different from other layers, you have:

• Hidden layer 1: 4 units (4 neurons)
• Hidden layer 2: 4 units
• Last layer: 1 unit

Shapes

Shapes are consequences of the model's configuration. Shapes are tuples representing how many elements an array or tensor has in each dimension.

Ex: a shape (30,4,10) means an array or tensor with 3 dimensions, containing 30 elements in the first dimension, 4 in the second and 10 in the third, totaling 30*4*10 = 1200 elements or numbers.

The input shape

What flows between layers are tensors. Tensors can be seen as matrices, with shapes.

In Keras, the input layer itself is not a layer, but a tensor. It's the starting tensor you send to the first hidden layer. This tensor must have the same shape as your training data.

Example: if you have 30 images of 50x50 pixels in RGB (3 channels), the shape of your input data is (30,50,50,3). Then your input layer tensor, must have this shape (see details in the "shapes in keras" section).

Each type of layer requires the input with a certain number of dimensions:

• Dense layers require inputs as (batch_size, input_size)
  • or (batch_size, optional,...,optional, input_size)
• 2D convolutional layers need inputs as:
  • if using channels_last: (batch_size, imageside1, imageside2, channels)
  • if using channels_first: (batch_size, channels, imageside1, imageside2)
• 1D convolutions and recurrent layers use (batch_size, sequence_length, features)
  • Details on how to prepare data for recurrent layers

Now, the input shape is the only one you must define, because your model cannot know it. Only you know that, based on your training data.

All the other shapes are calculated automatically based on the units and particularities of each layer.

Relation between shapes and units - The output shape

Given the input shape, all other shapes are results of layers calculations.

The "units" of each layer will define the output shape (the shape of the tensor that is produced by the layer and that will be the input of the next layer).

Each type of layer works in a particular way. Dense layers have output shape based on "units", convolutional layers have output shape based on "filters". But it's always based on some layer property. (See the documentation for what each layer outputs)

Let's show what happens with "Dense" layers, which is the type shown in your graph.

A dense layer has an output shape of (batch_size,units). So, yes, units, the property of the layer, also defines the output shape.

• Hidden layer 1: 4 units, output shape: (batch_size,4).
• Hidden layer 2: 4 units, output shape: (batch_size,4).
• Last layer: 1 unit, output shape: (batch_size,1).

Weights

Weights will be entirely automatically calculated based on the input and the output shapes. Again, each type of layer works in a certain way. But the weights will be a matrix capable of transforming the input shape into the output shape by some mathematical operation.

In a dense layer, weights multiply all inputs. It's a matrix with one column per input and one row per unit, but this is often not important for basic works.

In the image, if each arrow had a multiplication number on it, all numbers together would form the weight matrix.

Shapes in Keras

Earlier, I gave an example of 30 images, 50x50 pixels and 3 channels, having an input shape of (30,50,50,3).

Since the input shape is the only one you need to define, Keras will demand it in the first layer.

But in this definition, Keras ignores the first dimension, which is the batch size. Your model should be able to deal with any batch size, so you define only the other dimensions:

input_shape = (50,50,3)
    #regardless of how many images I have, each image has this shape        

Optionally, or when it's required by certain kinds of models, you can pass the shape containing the batch size via batch_input_shape=(30,50,50,3) or batch_shape=(30,50,50,3). This limits your training possibilities to this unique batch size, so it should be used only when really required.

Either way you choose, tensors in the model will have the batch dimension.

So, even if you used input_shape=(50,50,3), when keras sends you messages, or when you print the model summary, it will show (None,50,50,3).

The first dimension is the batch size, it's None because it can vary depending on how many examples you give for training. (If you defined the batch size explicitly, then the number you defined will appear instead of None)

Also, in advanced works, when you actually operate directly on the tensors (inside Lambda layers or in the loss function, for instance), the batch size dimension will be there.

• So, when defining the input shape, you ignore the batch size: input_shape=(50,50,3)
• When doing operations directly on tensors, the shape will be again (30,50,50,3)
• When keras sends you a message, the shape will be (None,50,50,3) or (30,50,50,3), depending on what type of message it sends you.

Dim

And in the end, what is dim?

If your input shape has only one dimension, you don't need to give it as a tuple, you give input_dim as a scalar number.

So, in your model, where your input layer has 3 elements, you can use any of these two:

• input_shape=(3,) -- The comma is necessary when you have only one dimension
• input_dim = 3

But when dealing directly with the tensors, often dim will refer to how many dimensions a tensor has. For instance a tensor with shape (25,10909) has 2 dimensions.

---

Defining your image in Keras

Keras has two ways of doing it, Sequential models, or the functional API Model. I don't like using the sequential model, later you will have to forget it anyway because you will want models with branches.

PS: here I ignored other aspects, such as activation functions.

With the Sequential model:

from keras.models import Sequential  
from keras.layers import *  

model = Sequential()    

#start from the first hidden layer, since the input is not actually a layer   
#but inform the shape of the input, with 3 elements.    
model.add(Dense(units=4,input_shape=(3,))) #hidden layer 1 with input

#further layers:    
model.add(Dense(units=4)) #hidden layer 2
model.add(Dense(units=1)) #output layer   

With the functional API Model:

from keras.models import Model   
from keras.layers import * 

#Start defining the input tensor:
inpTensor = Input((3,))   

#create the layers and pass them the input tensor to get the output tensor:    
hidden1Out = Dense(units=4)(inpTensor)    
hidden2Out = Dense(units=4)(hidden1Out)    
finalOut = Dense(units=1)(hidden2Out)   

#define the model's start and end points    
model = Model(inpTensor,finalOut)

Shapes of the tensors

Remember you ignore batch sizes when defining layers:

• inpTensor: (None,3)
• hidden1Out: (None,4)
• hidden2Out: (None,4)
• finalOut: (None,1)