Deep neural network models have been proven to be very successful in image classification tasks, also for medical diagnosis, but their main concern is its lack of interpretability. They use to work as intuition machines with high statistical confidence but unable to give interpretable explanations about the reported results. The vast amount of parameters of these models make difficult to infer a rationale interpretation from them. In this paper we present a diabetic retinopathy interpretable classifier able to classify retine images into the different levels of disease severity and of explaining its results by assigning a score for every point in the hidden and input space, evaluating its contribution to the final classification in a linear way. The generated visual maps can be interpreted by an expert in order to compare its own knowledge with the interpretation given by the model.
Another way for solving the interpretability problem is using the gradient of the classification function for the calculation of the next Taylor approximation @cite_7 :
@cite_7
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We consider the problem of determining functions of an image of an object that are insensitive to illumination changes. We first show that for an object with Lambertian reflectance there are no discriminative functions that are invariant to illumination. This result leads as to adopt a probabilistic approach in which we analytically determine a probability distribution for the image gradient as a function of the surface's geometry and reflectance. Our distribution reveals that the direction of the image gradient is insensitive to changes in illumination direction. We verify this empirically by constructing a distribution for the image gradient from more than 20 million samples of gradients in a database of 1,280 images of 20 inanimate objects taken under varying lighting condition. Using this distribution we develop an illumination insensitive measure of image comparison and test it on the problem of face recognition. In this paper we study the problem of minimizing the average of a large number ( @math ) of smooth convex loss functions. We propose a new method, S2GD (Semi-Stochastic Gradient Descent), which runs for one or several epochs in each of which a single full gradient and a random number of stochastic gradients is computed, following a geometric law. The total work needed for the method to output an @math -accurate solution in expectation, measured in the number of passes over data, or equivalently, in units equivalent to the computation of a single gradient of the loss, is @math , where @math is the condition number. This is achieved by running the method for @math epochs, with a single gradient evaluation and @math stochastic gradient evaluations in each. The SVRG method of Johnson and Zhang arises as a special case. If our method is limited to a single epoch only, it needs to evaluate at most @math stochastic gradients. In contrast, SVRG requires @math stochastic gradients. To illustrate our theoretical results, S2GD only needs the workload equivalent to about 2.1 full gradient evaluations to find an @math -accurate solution for a problem with @math and @math . Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on ℝN with smoothness of order s can in general be captured with accuracy at most O(n−s N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a⋅x) when both a∈ℝN and g∈C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g∈Cs[0,1]. We also study the role of sparsity or compressibility of a in such query problems. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.