Deeper networks should never have a higher **training** error than smaller ones. In the worst case, the layers should "simply" learn identities. It seems as this is not so easy with conventional networks, as they get much worse with more layers. So the idea is to add identity functions which skip some layers. The network only has to learn the **residuals**. 

Advantages:

* Learning the identity becomes learning 0 which is simpler
* Loss in information flow in the forward pass is not a problem anymore
    * No vanishing / exploding gradient
* Identities don't have parameters to be learned

## Evaluation

The learning rate starts at 0.1 and is divided by 10 when the error plateaus. Weight decay of 0.0001 ($10^{-4}$), momentum of 0.9. They use mini-batches of size 128.

* ImageNet ILSVRC 2015: 3.57% (ensemble)
* CIFAR-10: 6.43%
* MS COCO: 59.0% mAp@0.5 (ensemble)
* PASCAL VOC 2007: 85.6% mAp@0.5
* PASCAL VOC 2012: 83.8% mAp@0.5

## See also

* [DenseNets](http://www.shortscience.org/paper?bibtexKey=journals/corr/1608.06993)

FaceNet directly maps face images to $\mathbb{R}^{128}$ where distances directly correspond to a measure of face similarity. They use a triplet loss function. The triplet is (face of person A, other face of person A, face of person which is not A). Later, this is called (anchor, positive, negative).

The loss function is learned and inspired by LMNN. The idea is to minimize the distance between the two images of the same person and maximize the distance to the other persons image.

## LMNN

Large Margin Nearest Neighbor (LMNN) is learning a pseudo-metric

$$d(x, y) = (x -y) M  (x -y)^T$$

where $M$ is a positive-definite matrix. The only difference between a pseudo-metric and a metric is that $d(x, y) = 0 \Leftrightarrow x = y$ does not hold.

## Curriculum Learning: Triplet selection

Show simple examples first, then increase the difficulty. This is done by selecting the triplets.

They use the triplets which are *hard*. For the positive example, this means the distance between the anchor and the positive example is high. For the negative example this means the distance between the anchor and the negative example is low.

They want to have

$$||f(x_i^a) - f(x_i^p)||_2^2 + \alpha < ||f(x_i^a) - f(x_i^n)||_2^2$$

where $\alpha$ is a margin and $x_i^a$ is the anchor, $x_i^p$ is the positive face example and $x_i^n$ is the negative example. They increase $\alpha$ over time. It is crucial that $f$ maps the images not in the complete $\mathbb{R}^{128}$, but on the unit sphere. Otherwise one could double $\alpha$ by simply making $f' = 2 \cdot f$.

## Tasks

* **Face verification**: Is this the same person?
* **Face recognition**: Who is this person?

## Datasets

* 99.63% accuracy on Labeled FAces in the Wild (LFW)
* 95.12% accuracy on YouTube Faces DB

## Network

Two models are evaluated: The [Zeiler & Fergus model](http://www.shortscience.org/paper?bibtexKey=journals/corr/ZeilerF13)  and an architecture based on the [Inception model](http://www.shortscience.org/paper?bibtexKey=journals/corr/SzegedyLJSRAEVR14).

## See also

* [DeepFace](http://www.shortscience.org/paper?bibtexKey=conf/cvpr/TaigmanYRW14#martinthoma)

The main contribution of [Understanding the difficulty of training deep feedforward neural networks](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf) by Glorot et al. is a **normalized weight initialization**

$$W \sim U \left [  - \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}}, \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}} \right ]$$

where $n_j \in \mathbb{N}^+$ is the number of neurons in the layer $j$.

Showing some ways **how to debug neural networks** might be another reason to read the paper.

The paper analyzed standard multilayer perceptrons (MLPs) on a artificial dataset of $32 \text{px} \times 32 \text{px}$ images with either one or two of the 3 shapes: triangle, parallelogram and ellipse. The MLPs varied in the activation function which was used (either sigmoid, tanh or softsign).

However, no regularization was used and many mini-batch epochs were learned. It might be that batch normalization / dropout might change the influence of initialization very much.

Questions that remain open for me:

* [How is weight initialization done today?](https://www.reddit.com/r/MLQuestions/comments/4jsge9)
* Figure 4: Why is this plot not simply completely dependent on the data?
* Is softsign still used? Why not?
* If the only advantage of softsign is that is has the plateau later, why doesn't anybody use $\frac{1}{1+e^{-0.1 \cdot x}}$ or something similar instead of the standard sigmoid activation function?

We want to find two matrices $W$ and $H$ such that $V = WH$. Often a goal is to determine underlying patterns in the relationships between the concepts represented by each row and column. $W$ is some $m$ by $n$ matrix and we want the inner dimension of the factorization to be $r$. So 

$$\underbrace{V}_{m \times n} = \underbrace{W}_{m \times r} \underbrace{H}_{r \times n}$$

Let's consider an example matrix where of three customers (as rows) are associated with three movies (the columns) by a rating value.

$$
V = \left[\begin{array}{c c c}
5 & 4 & 1  \\\\
4 & 5 & 1 \\\\
2 & 1 & 5
\end{array}\right]
$$


We can decompose this into two matrices with $r = 1$. First lets do this without any non-negative constraint using an SVD reshaping matrices based on removing eigenvalues:


$$
W = \left[\begin{array}{c c c}
-0.656 \\\
 -0.652 \\\
 -0.379
\end{array}\right],
H = \left[\begin{array}{c c c}
-6.48 & -6.26 & -3.20\\\\
\end{array}\right]
$$

We can also decompose this into two matrices with $r = 1$ subject to the constraint that $w_{ij} \ge 0$ and  $h_{ij} \ge 0$. (Note: this is only possible when $v_{ij} \ge 0$):

$$
W = \left[\begin{array}{c c c}
0.388 \\\\
0.386 \\\\
0.224
\end{array}\right],
H = \left[\begin{array}{c c c}
11.22 & 10.57 & 5.41  \\\\
\end{array}\right]
$$

Both of these $r=1$ factorizations reconstruct matrix $V$ with the same error. 

$$
V \approx WH = \left[\begin{array}{c c c}
4.36 & 4.11 & 2.10 \\\
4.33 & 4.08 & 2.09 \\\
2.52 & 2.37 & 1.21 \\\
\end{array}\right]
$$


If they both yield the same reconstruction error then why is a non-negativity constraint useful? We can see above that it is easy to observe patterns in both factorizations such as similar customers and similar movies. `TODO: motivate why NMF is better`



#### Paper Contribution 

This paper discusses two approaches for iteratively creating a non-negative $W$ and $H$ based on random initial matrices. The paper discusses a multiplicative update rule where the elements of $W$ and $H$ are iteratively transformed by scaling each value such that error is not increased. 

The multiplicative approach is discussed in contrast to an additive gradient decent based approach where small corrections are iteratively applied. The multiplicative approach can be reduced to this by setting the learning rate ($\eta$) to a ratio that represents the magnitude of the element in $H$ to the scaling factor of $W$ on $H$.



### Still a draft

The paper introduces a sequential variational auto-encoder that generates complex images iteratively. The authors also introduce a new spatial attention mechanism that allows the model to focus on small subsets of the image. This new approach for image generation produces images that can’t be distinguished from the training data.

#### What is DRAW:
The deep recurrent attention writer (DRAW) model has two differences with respect to other variational auto-encoders. First, the encoder and the decoder are recurrent networks. Second, it includes an attention mechanism that restricts the input region observed by the encoder and the output region observed by the decoder.

#### What do we gain?
The resulting images are greatly improved by allowing a conditional and sequential generation. In addition, the spatial attention mechanism can be used in other contexts to solve the “Where to look?” problem.

#### What follows?
A possible extension to this model would be to use a convolutional architecture in the encoder or the decoder. Although this might be less useful since we are already restricting the input of the network.

#### Like:
* As observed in the samples generated by the model, the attention mechanism works effectively by reconstructing images in a local way.
* The attention model is fully differentiable.

#### Dislike:
* I think a better exposition of the attention mechanism would improve this paper.