Building Support Vector Machines in the Context of Regularised Least Squares

This paper formulates a linear kernel support vector machine (SVM) as a regularized least-squares (RLS) problem. By defining a set of indicator variables of the errors, the solution to the RLS problem is represented as an equation that relates the error vector to the indicator variables. Through partitioning the training set, the SVM weights and bias are expressed analytically using the support vectors. It is also shown how this approach naturally extends to Sums with nonlinear kernels whilst avoiding the need to make use of Lagrange multipliers and duality theory. A fast iterative solution algorithm based on Cholesky decomposition with permutation of the support vectors is suggested as a solution method. The properties of our SVM formulation are analyzed and compared with standard SVMs using a simple example that can be illustrated graphically. The correctness and behavior of our solution (merely derived in the primal context of RLS) is demonstrated using a set of public benchmarking problems for both linear and nonlinear SVMs.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Mental Health and Morbidity of Caregivers and Co-Residents of individuals with Dementia: A Quasi-Experimental Design

Objectives: To determine if providing informal care to a co-resident with dementia symptoms places an additional risk on the likelihood of poor mental health or mortality compared to co-resident non-caregivers.
Design: A quasi-experimental design of caregiving and non-caregiving co-residents of individuals with dementia symptoms, providing a natural comparator for the additive effects of caregiving on top of living with an individual with dementia symptoms. 
Methods: Census records, providing information on household structure, intensity of caregiving, presence of dementia symptoms and self-reported mental health, were linked to mortality records over the following 33 months. Multi-level regression models were constructed to determine the risk of poor mental health and death in co-resident caregivers of individuals with dementia symptoms compared to co-resident non-caregivers, adjusting for the clustering of individuals within households.
Results: The cohort consisted of 10,982 co-residents (55.1% caregivers), with 12.1% of non-caregivers reporting poor mental health compared to 8.4% of intense caregivers (>20 hours of care per week). During follow-up the cohort experienced 560 deaths (245 to caregivers). Overall, caregiving co-residents were at no greater risk of poor mental health but had lower mortality risk than non-caregiving co-residents (ORadj=0.93, 95% CI 0.79, 1.10 and ORadj=0.67, 95% CI 0.56, 0.81, respectively); this lower mortality risk was also seen amongst the most intensive caregivers (ORadj=0.65, 95% CI 0.53, 0.79).
Conclusion: Caregiving poses no additional risk to mental health over and above the risk associated with merely living with someone with dementia, and is associated with a lower mortality risk compared to non-caregiving co-residents.