Oblivious Sketching of High-Degree Polynomial Kernels

Kernel methods are fundamental tools in machine learning that allow detection of non-linear dependencies between data without explicitly constructing feature vectors in high dimensional spaces. A major disadvantage of kernel methods is their poor scalability: primitives such as kernel PCA or kernel ridge regression generally take prohibitively large quadratic space and (at least) quadratic time, as kernel matrices are usually dense. Some methods for speeding up kernel linear algebra are known, but they all invariably take time exponential in either the dimension of the input point set (e.g., fast multipole methods suffer from the curse of dimensionality) or in the degree of the kernel function. Oblivious sketching has emerged as a powerful approach to speeding up numerical linear algebra over the past decade, but our understanding of oblivious sketching solutions for kernel matrices has remained quite limited, suffering from the aforementioned exponential dependence on input parameters. Our main contribution is a general method for applying sketching solutions developed in numerical linear algebra over the past decade to a tensoring of data points without forming the tensoring explicitly. This leads to the first oblivious sketch for the polynomial kernel with a target dimension that is only polynomially dependent on the degree of the kernel function, as well as the first oblivious sketch for the Gaussian kernel on bounded datasets that does not suffer from an exponential dependence on the dimensionality of input data points

Cancer treatment: the combination of vaccination with other therapies

Harnessing of the immune system by the development of ‘therapeutic’ vaccines, for the battle against cancer has been the focus of tremendous research efforts over the past two decades. As an illustration of the impressive amounts of data gathered over the past years, numerous antigens expressed on the surface of cancer cells, have been characterized. To this end, recent years research has focussed on characterization of antigens that play an important role for the growth and survival of cancer cells. Anti-apoptotic molecules like survivin that enhance the survival of cancer cells and facilitate their escape from cytotoxic therapies represent prime vaccination candidates. The characterization of a high number of tumor antigens allow the concurrent or serial immunological targeting of different proteins associated with such cancer traits. Moreover, while vaccination in itself is a promising new approach to fight cancer, the combination with additional therapy could create a number of synergistic effects. Herein we discuss the possibilities and prospects of vaccination when combined with other treatments. In this regard, cell death upon drug exposure may be immunogenic or non-immunogenic depending on the specific chemotherapeutics. Also, chemotherapy represents one of several options available for clearance of CD4+ Foxp3+ regulatory T cells. Moreover, therapies based on monoclonal antibodies may have synergistic potential in combination with vaccination, both when used for targeting of tumor cells and endothelial cells. The efficacy of therapeutic vaccination against cancer will over the next few years be studied in settings taking advantage of strategies in which vaccination is combined with other treatment modalities. These combinations should be based on current knowledge not only regarding the biology of the cancer cell per se, but also considering how treatment may influence the malignant cell population as well as the immune system