kernel_computation(). The dual formulation of the SVM optimization problem introduces the notion of the kernel. In the case of linearly separable data, the kernel of two vectors is equivalent to the dot product of these two vectors. However, if we are dealing with non-separable data the kernel can be defined by a variety of functions. For instance, there is the polynomial kernel, the Gaussian kernel, the Laplacian kernel and others. The efficiency of a kernel function is determined by the nature of the training and tested input data, as well as other factors such as speed and accuracy. Hence, we selected a kernel function that was suitable for our training data set. In particular, we chose the exponential kernel function which according to the equivalent software implementation of this function is computed based on the following formula: In the above formula gamma denotes a constant selected by the user and dot produces the dot product of the given vectors. We have already described how the dot product can be performed with parallel computations. What is indicated by this formula is that all dot products can be computed in parallel too. For instance, if the dot product function receives 22-length vectors and all products are performed in parallel, then in total 22*3 products are produced in an instance. However, the result of dot(xi,xi) is a constant number as for every loop of the optimization solver i remains the same, and thereby we do not need to compute this dot product every time. It is the indicator j that changes during a loop - this indicator which receives all values from 0 up to the number of data instances of the input file. At this point we illustrate the specific loop, in order to provide better understanding. for(j=0;j<number_of_instances;j++) { result[j]=exp(-gamma*(x_i+dot(x[j],x[j])-2*dot(x[i],x[j],)); } , where x_i equals dot(xi,xi) and remains constant during the whole loop. Nevertheless, although the expression within the brackets can be implemented effectively in hardware, the same does not apply for the exponent. In particular, the computation of the exponent is expensive for hardware due to the fact that it occupies a significant number of resources and requires a noticeable amount of time to be executed. Thus, the entire formula will be calculated on the software side and only the expression within brackets will be implemented on hardware.
