## Terms

* Semantic Segmentation: Traditional segmentation divides the image in visually similar patches. Semantic segmentation on the other hand divides the image in semantically meaningful patches. This usually means to classify each pixel (e.g.: This pixel belongs to a cat, that pixel belongs to a dog, the other pixel is background).


## Main ideas

* Complete neural networks which were trained for image classification can be used as a convolution. Those networks can be trained on Image Net (e.g. VGG, AlexNet, GoogLeNet)
* Use upsampling to (1) reduce training and prediction time (2) improve consistency of output. (See [What are deconvolutional layers?](http://datascience.stackexchange.com/a/12110/8820) for an explanation.)


## How FCNs work

1. Train a neural network for image classification which is trained on input images of a fixed size ($d \times w \times h$)
2. Interpret the network as a single convolutional filter for each output neuron (so $k$ output neurons means you have $k$ filters) over the complete image area on which the original network was trained.
3. Run the network as a CNN over an image of any size (but at least $d \times w \times h$) with a stride $s \in \mathbb{N}_{\geq 1}$
4. If $s > 1$, then you need an upsampling layer (deconvolutional layer) to convert the coarse output into a dense output.

## Nice properties

* FCNs take images of arbitrary size and produce an image of the same output size.
* Computationally efficient

## See also:

https://www.quora.com/What-are-the-benefits-of-converting-a-fully-connected-layer-in-a-deep-neural-network-to-an-equivalent-convolutional-layer

> They allow you to treat the convolutional neural network as one giant filter. You can then spatially apply the neural net as a convolution to images larger than the original training image size, getting a spatially dense output.
>
> Let's say you train a neural net (with some loss function) with a convolutional layer (3 x 3, stride of 2), pooling layer (3 x 3, stride of 2), and a fully connected layer with 10 units, using 25 x 25 images. Note that the receptive field size of each max pooling unit is 7 x 7, so the pooling output is 5 x 5. You can convert the fully connected layer to to a set of  10 5 x 5 convolutional filters (unit strides). If you do that, the entire net can be treated as a filter with receptive field size 35 x 35 and stride of 4. You can then take that net and apply it to a 50 x 50 image, and you'd get a 3 x 3 x 10 spatially dense output.

This paper presents a linear algebraic trick for computing both the value and the gradient update for a loss function that compares a very high-dimensional target with a (dense) output prediction. Most of the paper exposes the specific case of the squared error loss, though it can also be applied to some other losses such as the so-called spherical softmax. One use case could be for training autoencoders with the squared error on very high-dimensional but sparse inputs. While a naive (i.e. what most people currently do) implementation would scale in $O(Dd)$ where $D$ is the input dimensionality and d the hidden layer dimensionality, they show that their trick allows to scale in $O(d^2)$. 

Their experiments show that they can achieve speedup factors of over 500 on the CPU, and over 1500 on the GPU. 

#### My two cents

This is a really neat, and frankly really surprising, mathematical contribution. I did not suspect getting rid of the dependence on D in the complexity would actually be achievable, even for the "simpler" case of the squared error. 

The jury is still out as to whether we can leverage the full power of this trick in practice. Indeed, the squared error over sparse targets isn't the most natural choice in most situations. The authors did try to use this trick in the context of a version of the neural network language model that uses the squared error instead of the negative log-softmax (or at least I think that's what was done... I couldn't confirm this with 100% confidence). They showed that good measures of word similarity (Simlex-999) could be achieved in this way, though using the hierarchical softmax actually achieves better performance in about the same time. 

But as far as I'm concerned, that doesn't make the trick less impressive. It's still a neat piece of new knowledge to have about reconstruction errors. Also, the authors mention that it would be possible to adapt the trick to the so-called (negative log) spherical softmax, which is like the softmax but where the numerator is the square of the pre-activation, instead of the exponential. I hope someone tries this out in the future, as perhaps it could be key to making this trick a real game changer!

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)

This paper deals with the question what / how exactly CNNs learn, considering the fact that they usually have more trainable parameters than data points on which they are trained.

When the authors write "deep neural networks", they are talking about Inception V3, AlexNet and MLPs.

## Key contributions

* Deep neural networks easily fit random labels (achieving a training error of 0 and a test error which is just randomly guessing labels as expected). $\Rightarrow$Those architectures can simply brute-force memorize the training data.
* Deep neural networks fit random images (e.g. Gaussian noise) with 0 training error. The authors conclude that VC-dimension / Rademacher complexity, and uniform stability are bad explanations for generalization capabilities of neural networks
* The authors give a construction for a 2-layer network with $p = 2n+d$ parameters - where $n$ is the number of samples and $d$ is the dimension of each sample - which can easily fit any labeling. (Finite sample expressivity). See section 4.


## What I learned

* Any measure $m$ of the generalization capability of classifiers $H$ should take the percentage of corrupted labels ($p_c \in [0, 1]$, where $p_c =0$ is a perfect labeling and $p_c=1$ is totally random) into account: If $p_c = 1$, then $m()$ should be 0, too, as it is impossible to learn something meaningful with totally random labels.
* We seem to have built models which work well on image data in general, but not "natural" / meaningful images as we thought.


## Funny

> deep neural nets remain mysterious for many reasons

> Note that this is not exactly simple as the kernel matrix requires 30GB to store in memory. Nonetheless, this system can be solved in under 3 minutes in on a commodity workstation with 24 cores and 256 GB of RAM with a conventional LAPACK call.


## See also

* [Deep Nets Don't Learn Via Memorization](https://openreview.net/pdf?id=rJv6ZgHYg)

Liang et al. propose a perturbation-based approach for detecting out-of-distribution examples using a network’s confidence predictions. In particular, the approaches based on the observation that neural network’s make more confident predictions on images from the original data distribution, in-distribution examples, than on examples taken from a different distribution (i.e., a different dataset), out-distribution examples. This effect can further be amplified by using a temperature-scaled softmax, i.e.,

$ S_i(x, T) = \frac{\exp(f_i(x)/T)}{\sum_{j = 1}^N \exp(f_j(x)/T)}$

where $f_i(x)$ are the predicted logits and $T$ a temperature parameter. Based on these softmax scores, perturbations $\tilde{x}$ are computed using

$\tilde{x} = x - \epsilon \text{sign}(-\nabla_x \log S_{\hat{y}}(x;T))$

where $\hat{y}$ is the predicted label of $x$. This is similar to “one-step” adversarial examples; however, in contrast of minimizing the confidence of the true label, the confidence in the predicted label is maximized. This, applied to in-distribution and out-distribution examples is illustrated in Figure 1 and meant to emphasize the difference in confidence. Afterwards, in- and out-distribution examples can be distinguished using simple thresholding on the predicted confidence, as shown in various experiment, e.g., on Cifar10 and Cifar100.

https://i.imgur.com/OjDVZ0B.png
Figure 1: Illustration of the proposed perturbation to amplify the difference in confidence between in- and out-distribution examples.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).