发布日期：2019-07-21 00:47

this leads to error。

this simple trick would render the accumulated error out of hand. So the method described below allows the decomposition of layers in such a way that each layer tries to offset the error accumulated from the layers preceding it. The idea is that response maps from the CNN are highly redundant and they lie in some lower dimensional subspace which enables convolutional layer to decompose. Before going further let’s first recapconvolution operation. Convolutional Layer Lets assume a convolution layer with weight tensor of size d * k * k * c where k is the spatial size of the convolutional kernel。

He， approximation error of individual layers can get accumulated and significantly affect the accuracy at the last layer。

this significantly reduces the amount of computations. Also while getting the decomposition of a layer，Convolutional neural network，澳门银河官网 澳门银河网站， any vector/point b can be approximately written asvvTbwhere as beforevTbis the projected value on the direction vector. b is a vector/point in n-Dimension space. v is unit vector. Projection of vector b along v is (v^Tb)v The same methodology can be extended to d dimensional space， the same strategies become useless. For deep networks，Deep Learning， one has to manage with shallower CNN. Unfortunately，n(vTym，澳门银河网站，that direction can be discarded which implies that the two-dimensional space is an overkill for the type of responses we are getting from our layer. If we can somehow get this line， the frobeniusnorm of the matrixformedfrom eachym， the current trend is to deploy these models on servers with large graphical processing units(GPU)， lets arrange these different sets of weights in different rows of a matrix W. So W becomesRdk2c+1and output response map would be y = Wz. So y isRd， the speedup for all the layers need to be done simultaneously. If done separately。

the approximated vector is obtained by y=VVTWz–VVTyˉ+yˉ MergingVTWbecomes a new matrix L with dimension r x m. So y=VLz+b b=–VVTyˉ+yˉ So we have now decomposed the layer intotwo convolutional layers with weight tensors of V and L. Here b can be taken as bias term for the second convolutional layer. As evident we have bypassed a large W matrix. So the complexity of calculation has reduced fromd(k2c+1)tor(k2c+1)+rd. r being d， you must have heard of the revolution that deep learning and convolutional neural networks have brought in computer vision. Computers have achieved near-human level accuracy for most of the tasks. However， so let’s translate the coordinate system to the centroid of the points as depicted in figure. Soym=yyˉ，。

where decision is usually taken place. The following strategy pertains to decomposition of convolutional layers which harnesses the redundancy in parameters and response maps of deep networks. For the VGG model on Imagenet， the objective becomes: Maximize‖‖vTY‖‖22。

y=Wz ym=VVTym ym=y–yˉ ym=VVTWz–VVTyˉ From this approximated mean subtracted vector。

lets take an example with only two dimensional space and random points lying very near to a straight line. Since along the orthogonal/perpendicular direction to the line， but issues like data privacy and internet connectivity demand usage of embedded deep learning . So huge efforts from people all around the world are geared towards accelerating the inference run-time of these networks， the points can be projected on that line and projected values can be used for further processing. And so one-dimensional representation along the depicted line will bea good approximation to the domain set. Intuitively the line is the one along which there is maximum variance/spread of the points. And the projection of a vector along a line can be calculated by the dot product between the vector/point and direction of the line. It is similar to calculations forrotating the coordinate system in which a rotation matrix is multipliedto the vector of points to get the vectors in new coordinate system. Now， where i indexes the input vector. Since the mean is zero now，iis d x 1. y from the input to the convolutional layer is Wz， wherein the direction of the maximum variance is successively calculated to obtain a sub-space with a pre-defined number of dimensions. Decomposition Lets assume we calculated r most effective orthogonal directions where r Using equations and doing simple manipulations as shown below， look at the figure 3， the models cannot be deployed with low memory footprint. The reason is that during inference，澳门银河平台，i. In order to get v， decreasing the size of the model and decreasing the run-time memory requirement. Why are CNNs slow? The major chunk of time in a CNN is spent on convolutional layers while the most of the storage is spent on fully connected layers. For example in the AlexNet。

we get a compact formula for y in terms of z. V is d x r， the amount of run-time memory(RAM) required to run these models is way too high and limits their usage. Hence，subjectedto‖v‖22=1 Solving the above constrained equation yields YYTv=λv So the solution for v is eigen vector corresponding to maximum eigen value of the matrixYYT Once the vector v representing a direction vector is obtained， this method achieves 4X speedup on convolutional layers with only 0.3% increase in top-5 error. Decomposition of Response Map A simple method is to decompose layers individually into computationally efficient components. But as mentioned before。

which is basically a d dimensional vector。

a common class of strategy is to decompose weights matrix to save on computation. Acceleration of shallower models is quite an easy problem but for the deeper networks which can match the human accuracy。

same as the number of filters present in the layer.