Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems

We consider conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle-point problems or, in particular, regularizations thereof. Such methods require preconditioners that preserve certain sub-blocks from the original systems but allow considerable flexibility for the remaining blocks. We construct a number of families of implicit factorizations that are capable of reproducing the required sub-blocks and (some) of the remainder. These generalize known implicit factorizations for the unregularized case. Improved eigenvalue clustering is possible if additionally some of the noncrucial blocks are reproduced. Numerical experiments confirm that these implicit-factorization preconditioners can be very effective in practice.

Fast and frugal framing effects?

Three experiments examine whether simple pair-wise comparison judgments, involving the “recognition heuristic” (Goldstein & Gigerenzer, 2002), are sensitive to implicit cues to the nature of the comparison required. Experiments 1 & 2 show that participants frequently choose the recognized option of a pair if asked to make “larger” judgments but are significantly less likely to choose the unrecognized option when asked to make “smaller” judgments. Experiment 3 demonstrates that, overall, participants consider recognition to be a more reliable guide to judgments of a magnitude criterion than lack of recognition and that this intuition drives the framing effect. These results support the idea that, when making pair-wise comparison judgments, inferring that the recognized item is large is simpler than inferring that the unrecognized item is small.

Less-is-more effects without the recognition heuristic

Inferences consistent with “recognition-based” decision-making may be drawn for various reasons other than recognition alone. We demonstrate that, for 2-alternative forced-choice decision tasks, less-is-more effects (reduced performance with additional learning) are not restricted to recognition-based inference but can also be seen in circumstances where inference is knowledge-based but item knowledge is limited. One reason why such effects may not be observed more widely is the dependence of the effect on specific values for the validity of recognition and knowledge cues. We show that both recognition and knowledge validity may vary as a function of the number of items recognized. The implications of these findings for the special nature of recognition information, and for the investigation of recognition-based inference, are discussed

Implicit synthesis

In this reply to Neal and Hesketh and to the commentators, we argue that implicit knowledge is partly abstract and can be usefully defined by the criteria of both metaknowledge and intentional control. We suggest that the pattern of dissociations supports a claim of separate implicit and explicit learning modes. According to our characterization, implicit learning leads to knowledge that is not automatically represented as knowledge by the learning process; instead, the presence of knowledge has to be inferred by the subject (partial explicitation) if metaknowledge is gained at all. During explicit learning, knowledge is automatically labeled as knowledge by the learning process, so that metaknowledge comes immediately and to the fullest extent. Finally, we suggest that implicit knowledge may to some degree apply regardless of intention.