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 derives an algorithm for passing gradients through a sample from a mixture of Gaussians. While the reparameterization trick allows to get the gradients with respect to the Gaussian means and covariances, the same trick cannot be invoked for the mixing proportions parameters (essentially because they are the parameters of a multinomial discrete distribution over the Gaussian components, and the reparameterization trick doesn't extend to discrete distributions).

One can think of the derivation as proceeding in 3 steps:

1. Deriving an estimator for gradients a sample from a 1-dimensional density $f(x)$ that is such that $f(x)$ is differentiable and its cumulative distribution function (CDF) $F(x)$ is tractable:

  $\frac{\partial \hat{x}}{\partial \theta} = - \frac{1}{f(\hat{x})}\int_{t=-\infty}^{\hat{x}} \frac{\partial f(t)}{\partial \theta} dt$

  where $\hat{x}$ is a sample from density $f(x)$ and $\theta$ is any parameter of $f(x)$ (the above is a simplified version of Equation 6). This is probably the most important result of the paper, and is based on a really clever use of the general form of the Leibniz integral rule.

2. Noticing that one can sample from a $D$-dimensional Gaussian mixture by decomposing it with the product rule $f({\bf x}) = \prod_{d=1}^D f(x_d|{\bf x}_{<d})$ and using ancestral sampling, where each $f(x_d|{\bf x}_{<d})$ are themselves 1-dimensional mixtures (i.e. with differentiable densities and tractable CDFs)

3. Using the 1-dimensional gradient estimator (of Equation 6) and the chain rule to backpropagate through the ancestral sampling procedure. This requires computing the integral in the expression for $\frac{\partial \hat{x}}{\partial \theta}$ above, where $f(x)$ is one of the 1D conditional Gaussian mixtures and $\theta$ is a mixing proportion parameter $\pi_j$. As it turns out, this integral has an analytical form (see Equation 22).

**My two cents**

This is a really surprising and neat result. The author mentions it could be applicable to variational autoencoders (to support posteriors that are mixtures of Gaussians), and I'm really looking forward to read about whether that can be successfully done in practice. 

The paper provides the derivation only for mixtures of Gaussians with diagonal covariance matrices. It is mentioned that extending to non-diagonal covariances is doable. That said, ancestral sampling with non-diagonal covariances would become more computationally expensive, since the conditionals under each Gaussian involves a matrix inverse.

Beyond the case of Gaussian mixtures, Equation 6 is super interesting in itself as its application could go beyond that case. This is probably why the paper also derived a sampling-based estimator for Equation 6, in Equation 9. However, that estimator might be inefficient, since it involves sampling from Equation 10 with rejection, and it might take a lot of time to get an accepted sample if $\hat{x}$ is very small. Also, a good estimate of Equation 6 might require *multiple* samples from Equation 10. 

Finally, while I couldn't find any obvious problem with the mathematical derivation, I'd be curious to see whether using the same approach to derive a gradient on one of the Gaussian mean or standard deviation parameters gave a gradient that is consistent with what the reparameterization trick provides.

Mask RCNN takes off from where Faster RCNN left, with some augmentations aimed at bettering instance segmentation (which was out of scope for FRCNN). Instance segmentation was achieved remarkably well in *DeepMask* , *SharpMask* and later *Feature Pyramid Networks* (FPN).

Faster RCNN was not designed for pixel-to-pixel alignment between network inputs and outputs. This is most evident in how RoIPool , the de facto core operation for attending to instances, performs coarse spatial quantization for feature extraction. Mask RCNN fixes that by introducing RoIAlign in place of RoIPool.

#### Methodology

Mask RCNN retains most of the architecture of Faster RCNN. It adds the a third branch for segmentation. The third branch takes the output from RoIAlign layer and predicts binary class masks for each class.

##### Major Changes and intutions

**Mask prediction**

Mask prediction segmentation predicts a binary mask for each RoI using fully convolution - and the stark difference being usage of *sigmoid* activation for predicting final mask instead of *softmax*, implies masks don't compete with each other. This *decouples* segmentation from classification. The class prediction branch is used for class prediction and for calculating loss, the mask of predicted loss is used calculating Lmask.

Also, they show that a single class agnostic mask prediction works almost as effective as separate mask for each class, thereby supporting their method of decoupling classification from segmentation

**RoIAlign**

RoIPool first quantizes a floating-number RoI to the discrete granularity of the feature map, this quantized RoI is then subdivided into spatial bins which are themselves quantized, and finally feature values covered by each bin are aggregated (usually by max pooling). Instead of  quantization of the RoI boundaries
or bin bilinear interpolation is used to compute the exact values of the input features at four regularly sampled locations in each RoI bin, and aggregate the result (using max or average).

**Backbone architecture**

Faster RCNN uses a VGG like structure for extracting features from image, weights of which were shared among RPN and region detection layers. Herein, authors experiment with 2 backbone architectures - ResNet based VGG like in FRCNN and ResNet based [FPN](http://www.shortscience.org/paper?bibtexKey=journals/corr/LinDGHHB16) based. FPN uses convolution feature maps from previous layers and recombining them to produce pyramid of feature maps to be used for prediction instead of single-scale feature layer (final output of conv layer before connecting to fc layers was used in Faster RCNN) 

**Training Objective**

The training objective looks like this 
![](https://i.imgur.com/snUq73Q.png)

Lmask is the addition from Faster RCNN. The method to calculate was mentioned above

#### Observation

Mask RCNN performs significantly better than COCO instance segmentation winners *without any bells and whiskers*. Detailed results are available in the paper

Ilyas et al. present a follow-up work to their paper on the trade-off between accuracy and robustness. Specifically, given a feature $f(x)$ computed from input $x$, the feature is considered predictive if

$\mathbb{E}_{(x,y) \sim \mathcal{D}}[y f(x)] \geq \rho$;

similarly, a predictive feature is robust if

$\mathbb{E}_{(x,y) \sim \mathcal{D}}\left[\inf_{\delta \in \Delta(x)} yf(x + \delta)\right] \geq \gamma$.

This means, a feature is considered robust if the worst-case correlation with the label exceeds some threshold $\gamma$; here the worst-case is considered within a pre-defined set of allowed perturbations $\Delta(x)$ relative to the input $x$. Obviously, there also exist predictive features, which are however not robust according to the above definition. In the paper, Ilyas et al. present two simple algorithms for obtaining adapted datasets which contain only robust or only non-robust features. The main idea of these algorithms is that an adversarially trained model only utilizes robust features, while a standard model utilizes both robust and non-robust features. Based on these datasets, they show that non-robust, predictive features are sufficient to obtain high accuracy; similarly training a normal model on a robust dataset also leads to reasonable accuracy but also increases robustness. Experiments were done on Cifar10. These observations are supported by a theoretical toy dataset consisting of two overlapping Gaussians; I refer to the paper for details.

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

This paper presents a novel neural network approach (though see [here](https://www.facebook.com/hugo.larochelle.35/posts/172841743130126?pnref=story) for a discussion on prior work) to density estimation, with a focus on image modeling. At its core, it exploits the following property on the densities of random variables. Let $x$ and $z$ be two random variables of equal dimensionality such that $x = g(z)$, where $g$ is some bijective and deterministic function (we'll note its inverse as $f = g^{-1}$). Then the change of variable formula gives us this relationship between the densities of $x$ and $z$:

$p_X(x) = p_Z(z) \left|{\rm det}\left(\frac{\partial g(z)}{\partial z}\right)\right|^{-1}$

Moreover, since the determinant of the Jacobian matrix of the inverse $f$ of a function $g$ is simply the inverse of the Jacobian of the function $g$, we can also write:

$p_X(x) = p_Z(f(x)) \left|{\rm det}\left(\frac{\partial f(x)}{\partial x}\right)\right|$

where we've replaced $z$ by its deterministically inferred value $f(x)$ from $x$.

So, the core of the proposed model is in proposing a design for bijective functions $g$ (actually, they design its inverse $f$, from which $g$ can be derived by inversion), that have the properties of being easily invertible and having an easy-to-compute determinant of Jacobian. Specifically, the authors propose to construct $f$ from various modules that all preserve these properties and allows to construct highly non-linear $f$ functions. Then, assuming a simple choice for the density $p_Z$ (they use a multidimensional Gaussian), it becomes possible to both compute $p_X(x)$ tractably and to sample from that density, by first samples $z\sim p_Z$ and then computing $x=g(z)$.

The building blocks for constructing $f$ are the following:

**Coupling layers**: This is perhaps the most important piece. It simply computes as its output $b\odot x + (1-b) \odot (x \odot \exp(l(b\odot x)) + m(b\odot x))$, where $b$ is a binary mask (with half of its values set to 0 and the others to 1) over the input of the layer $x$, while $l$ and $m$ are arbitrarily complex neural networks with input and output layers of equal dimensionality. 

In brief, for dimensions for which $b_i = 1$ it simply copies the input value into the output. As for the other dimensions (for which $b_i = 0$) it linearly transforms them as $x_i * \exp(l(b\odot x)_i) + m(b\odot x)_i$. Crucially, the bias ($m(b\odot x)_i$) and coefficient ($\exp(l(b\odot x)_i)$) of the linear transformation are non-linear transformations (i.e. the output of neural networks) that only have access to the masked input (i.e. the non-transformed dimensions). While this layer might seem odd, it has the important property that it is invertible and the determinant of its Jacobian is simply $\exp(\sum_i (1-b_i) l(b\odot x)_i)$. See Section 3.3 for more details on that.

**Alternating masks**: One important property of coupling layers is that they can be stacked (i.e. composed), and the resulting composition is still a bijection and is invertible (since each layer is individually a bijection) and has a tractable determinant for its Jacobian (since the Jacobian of the composition of functions is simply the multiplication of each function's Jacobian matrix, and the determinant of the product of square matrices is the product of the determinant of each matrix). This is also true, even if the mask $b$ of each layer is different. Thus, the authors propose using masks that alternate across layer, by masking a different subset of (half of) the dimensions. For images, they propose using masks with a checkerboard pattern (see Figure 3). Intuitively, alternating masks are better because then after at least 2 layers, all dimensions have been transformed at least once.

**Squeezing operations**: Squeezing operations corresponds to a reorganization of a 2D spatial layout of dimensions into 4 sets of features maps with spatial resolutions reduced by half (see Figure 3). This allows to expose multiple scales of resolutions to the model. Moreover, after a squeezing operation, instead of using a checkerboard pattern for masking, the authors propose to use a per channel masking pattern, so that "the resulting partitioning is not redundant with the previous checkerboard masking". See Figure 3 for an illustration.

Overall, the models used in the experiments usually stack a few of the following "chunks" of layers: 1) a few coupling layers with alternating checkboard masks, 2) followed by squeezing, 3) followed by a few coupling layers with alternating channel-wise masks. Since the output of each layers-chunk must technically be of the same size as the input image, this could become expensive in terms of computations and space when using a lot of layers. Thus, the authors propose to explicitly pass on (copy) to the very last layer ($z$) half of the dimensions after each layers-chunk, adding another chunk of layers only on the other half. This is illustrated in Figure 4b.

Experiments on CIFAR-10, and 32x32 and 64x64 versions of ImageNet show that the proposed model (coined the real-valued non-volume preserving or Real NVP) has competitive performance (in bits per dimension), though slightly worse than the Pixel RNN.

**My Two Cents**

The proposed approach is quite unique and thought provoking. Most interestingly, it is the only powerful generative model I know that combines A) a tractable likelihood, B) an efficient / one-pass sampling procedure and C) the explicit learning of a latent representation. While achieving this required a model definition that is somewhat unintuitive, it is nonetheless mathematically really beautiful!

I wonder to what extent Real NVP is penalized in its results by the fact that it models pixels as real-valued observations. First, it implies that its estimate of bits/dimensions is an upper bound on what it could be if the uniform sub-pixel noise was integrated out (see Equations 3-4-5 of [this paper](http://arxiv.org/pdf/1511.01844v3.pdf)). Moreover, the authors had to apply a non-linear transformation (${\rm logit}(\alpha + (1-\alpha)\odot x)$) to the pixels, to spread the $[0,255]$ interval further over the reals. Since the Pixel RNN models pixels as discrete observations directly, the Real NVP might be at a disadvantage.

I'm also curious to know how easy it would be to do conditional inference with the Real NVP. One could imagine doing approximate MAP conditional inference, by clamping the observed dimensions and doing gradient descent on the log-likelihood with respect to the value of remaining dimensions. This could be interesting for image completion, or for structured output prediction with real-valued outputs in general. I also wonder how expensive that would be.

In all cases, I'm looking forward to saying interesting applications and variations of this model in the future!