## Description

This assignment introduces you to some theoretical results that address which neural networks can

represent what. It walks you through proving some simplified versions of more general results. In

particular, this assignment focuses on piecewise linear neural networks, which are the most common

type at the moment. The general strategy will be to construct a neural network that has the desired

properties by choosing appropriate sets of weights.

1 Convolution Basics

The convolution layer inside of a CNN is intrinsically an affine transformation: A vector is received

as input and is multiplied with a matrix to produce an output (to which a bias vector is usually

added before passing the result through a nonlinearity). This operation can be represented as

y = Ax, in which A describes the affine transformation.

1. [2 points] We will first revisit the convolution layer as discussed in the class. Consider a

convolution layer with a 3×3 kernel W, operated on a single input channel X, represented as:

W =

w(0,0) w(0,1) w(0,2)

w(1,0) w(1,1) w(1,2)

w(2,0) w(2,1) w(2,2)

, X =

x(0,0) x(0,1) x(0,2)

x(1,0) x(1,1) x(1,2)

x(2,0) x(2,1) x(2,2)

(1)

Now let us work out a stride-2 convolution layer, with zero-padding size of 1. Consider

‘flattening’ the input tensor X in row-major order as:

X =

x(0,0), x(0,1), x(0,2), …, x(2,0), x(2,1), x(2,2)>

(2)

Write down the convolution as a matrix operation A that: Y = AX. Output Y is also

flattened in row-major order.

2. [2 points] Recall that transposed convolution can help us upsample the feature size spatially.

Consider a transposed convolution layer with a 2×2 kernel W operated on a single input

channel X, represented as:

W =

w(0,0), w(0,1)

w(1,0), w(1,1)

, X =

x(0,0), x(0,1)

x(1,0), x(1,1)

(3)

We ‘flattened’ the input tensor in row-major order as X = [x(0,0), x(0,1), x(1,0), x(1,1)].

Write down the affine transformation A corresponding to a transposed convolution layer with

kernel W, stride 2, no padding. Output Y is also flattened in row-major order.

2 Logic

1. [4 points] Implement AND and OR for pairs of binary inputs using a single linear threshold

neuron with weights w ∈ R

2

, bias b ∈ R, and x ∈ {0, 1}

2

:

f(x) = (

1 if wT x + b ≥ 0

0 if wT x + b < 0

(4)

2

That is, find wAND and bAND such that

x1 x2 fAND(x)

0 0 0

0 1 0

1 0 0

1 1 1

Also find wOR and bOR such that

x1 x2 fOR(x)

0 0 0

0 1 1

1 0 1

1 1 1

2. [4 points] Consider the bidirectional implication “⇔” (i.e., “if and only if”, often abbreviated

as “iff”), which has the following truth table

x1 x2 f⇔(x)

0 0 1

0 1 0

1 0 0

1 1 1

Show that “⇔” can NOT be represented using a linear model with the same form as (4).

[Hint: To see why, plot the examples from above in a plane and think about drawing a linear

boundary that separates them.]

3 Depth – Composing Linear Pieces [Extra Credit for 4803; Regular

Credit for 7643]

An interesting property of networks with piecewise linear activations like the ReLU is that on the

whole they compute piecewise linear functions. For instance, each layer h

(i) below computes a linear

function of the previous layer, h

(i−1), and the resulting function is piecewise-linear in x:

h

(1)(x) = W(1)x + b

(1) (5)

h

(2)(x) = W(2) max{h

(1)(x), 0} + b

(2) (6)

h

(3)(x) = W(3) max{h

(2)(x), 0} + b

(3) (7)

Now we’ll turn to a more recent result that highlights the Deep in Deep Learning. Depth (composing more functions) results in a favorable combinatorial explosion in the “number of things that a

neural net can represent”. For example, to classify a cat it seems useful to first find parts of a cat:

eyes, ears, tail, fur, etc. The function which computes a probability of cat presence should be a

function of these components because this allows everything you learn about eyes to generalize to

all instances of eyes instead of just a single instance. Below you will detail one formalizable sense

of this combinatorial explosion for a particular class of piecewise linear networks.

3

Figure 1

Consider y = σ(x) = |x| for scalar x ∈ X ⊆ R and y ∈ Y ⊆ R (Fig. 1). It has one linear region on x < 0 and another on x > 0 and the activation identifies these regions, mapping both of

them to y > 0. More precisely, for each linear region of the input, σ(·) is a bijection. There is

a mapping to and from the output space and the corresponding input space. However, given an

output y, it’s impossible to tell which linear region of the input it came from, thus σ(·) identifies

(maps on top of each other) the two linear regions of its input. This is the crucial definition because

when a function identifies multiple regions of its domain that means any subsequent computation

applies to all of those regions. When these regions come from an input space like the space of images, functions which identify many regions where different images might fall (e.g., slightly different

images of a cat) automatically transfer what they learn about a particular cat to cats in the other

regions.

More formally, we will say that σ(·) identifies a set of M disjoint input regions R = {R1, . . . , RM}

(e.g., R = {(−1, 0),(0, 1)}) with Ri ⊆ X onto one output region O ⊆ Y (e.g., (0, 1)) if for all

Ri ∈ R there is a bijection from Ri to O.

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1. [4 points] Start by applying the above notion of identified regions to linear regions of one

layer of a particular neural net that uses absolute value functions as activations. Let x ∈ R

d

,

y ∈ R

d 2

, and pick weights W(1) ∈ R

d×d and bias b

(1) ∈ R

d as follows:

W

(1)

ij =

(

2 if i = j

0 if i 6= j

(8)

b

(1)

i = −1 (9)

Then one layer of a neural net with absolute value activation functions is given by

f1(x) = |W(1)x + b| (10)

Note that this is an absolute value function applied piecewise and not a norm.

How many regions of the input are identified onto O = (0, 1)d by f1(·)? Prove it. 3

1Recall that a bijection from X to Y is a function µ : X → Y such that for all y ∈ Y there exists a unique x ∈ X

with µ(x) = y.

2Outputs are in some feature space, not a label space. Normally a linear classifier would be placed on top of what

we are here calling y.

3Absolute value activations are chosen to make the problem simpler, but a similar result holds for ReLU units.

Also, O could be the positive orthant (unbounded above).

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2. [4 points] Next consider what happens when two of these functions are composed. Suppose

g identifies ng regions of (0, 1)d onto (0, 1)d and f identifies nf regions of (0, 1)d onto (0, 1)d

.

How many regions of its input does f ◦ g(·) identify onto (0, 1)d

?

3. [6 points] Finally consider a series of L layers identical to the one in question 4.1.

h1 = |W1x + b1| (11)

h2 = |W2h1 + b2| (12)

.

.

. (13)

hL = |WLhL−1 + bL| (14)

Let x ∈ (0, 1)d and f(x) = hL. Note that each hi

is implicitly a function of x. Show that

f(x) identifies 2

Ld regions of its input.

4 Conclusion to Theory Part

Now compare the number of identified regions for an L layer net to that of an L − 1 layer net. The

L layer net can separate its input space into 2

d more linear regions than the L − 1 layer net. On

the other hand, the number of parameters and the amount of computation time grows linearly in

the number of layers. In this very particular sense (which doesn’t always align well with practice)

deeper is better.

To summarize this problem set, you’ve shown a number of results about the representation power

of different neural net architectures. First, neural nets (even single neurons) can represent logical

operations. Second, neural nets we use today compute piecewise linear functions of their input.

Third, the representation power of neural nets increases exponentially with the number of layers.

The point of the exercise was to convey intuition that removes some of the magic from neural nets

representations. Specifically, neural nets can decompose problems logically, and “simple” piecewise

linear functions can be surprisingly powerful.

5 Paper Review

The second of our paper reviews for this course comes from some interesting work at Facebook AI

Research (FAIR) last year exploring the limits of weakly supervised learning and improving upon

their results on the ImageNet dataset by appending data from Instagram using hashtags as ground

truth labels. The paper can be viewed here.

Guidelines: Please restrict your answers to no more than 350 words combined. The evaluation

rubric for this section is as follows :

1. [2 points] The paper shows that training with very weak (noisy) labels still leads to robust

learning of features that generalize well. Why is deep learning so robust to noise? Provide

some conjectures, relating it to what you know about machine/deep learning and optimization.

2. [2 points] Another finding is that learning features using similar label spaces (i.e. on categories that overlap with what you are hoping to generalize to) is more successful than learning

features on dissimilar label spaces. Why might that be the case?

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6 Coding: Gradients With Respect to Input [65 points]

The coding part of this assignment will have you implement different ideas which all rely on

gradients of some neural net output with respect to the input image. To get started go to

https://www.cc.gatech.edu/classes/AY2020/cs7643_spring/Z3o9P26CwTPZZMDXyWYDj3/hw2/.

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