In the past, I have published in diverse and broad areas of
§ I started off studying how hyperacuity may arise in sensory representation and how resolution and magnification of cortical maps are inversely related through dynamic self-organization models. I then devote significant effort into understanding motion perception, in particular motion-based figure-ground segregation (as evidenced in random-dot kinematogram). Following up researches on the then-hot “aperture problem”, I refined (with correct mathematical formulation) a method for separating orientation and direction tuning in visual neuron’s response to oriented drifting stimuli, explained an apparent puzzle (by mathematical modeling) in directional selective neurons’ response to drifting random-dot pattern, and demonstrated (via psychophysics experiments) that the “softness” of an aperture determines whether motion integration vs motion contrast is to occur.
§ The highlight of these vision-related research (dating back to my Ph.D. years) is a mathematical model on (motion-based) perceptual binding. Using the tools of differentiable manifold, I proposed that figure-ground segregation and object “oneness”, in terms of computation, can be characterized as a constant vector field (under Levi-Civita connection), where vectors at each point of the manifold (2-D visual space) represent the measurements of motion sensors that are susceptible to the aperture problem. Conceptualizing visual perception as having the (mathematical) structure of fiber bundles, with base manifold being the 2-D frontal-parallel visual space, has many advantages that is my continued research interest nowadays.
· Theory of Signal Detection is, in offering a theoretical framework for separating sensitivity and decision bias aspects of simple detection and discrimination tasks. The “non-parametric” estimate of sensitivity and bias factors is a popular alternative to parametric estimate (based on the Gaussian model); yet I found that the formula for calculating the so-called A’ index, and even a purported correction of it, contains an error. I provided (with computer graphical help from a student Shane Mueller) the definitive formula for calculating what most people believe it should calculate, and have recently extended to the case when experimental manipulations yield a second point in the ROC space.
· I have looked into the conceptualization of stimulus-response compatibility by Sylvan Kornblum, and collaborated with him and his associates (including his student Huazhong Zhang) on developing a connectionist model of S-R compatibility effects. A by-product is a cute mathematical proof that a linear plot in the so-called “distributional analysis” of two RT distributions actually reveals that they are members of a location-scale family (contrary to what the analysis was intended to reveal).
· Since 2007, I have developed a strong interest in understanding neural processes that mediate sensor-motor “decisions”. The experimental setting is that of trial-by-trial recording of neural activities (single neuron, ERP, fMRI, etc) while a behaving animal or human is carrying out a simple SR task. I advanced three methods, that can be used in orthogonal fashions, to determine a recorded neural response (single neuron, ERP, fMRI, etc) should be considered to related to the online processing of stimulus, response, or the decision that mediate the two. First, a TSD-based index for sensorimotor “locus” of a neuron can be calculated, when sufficient error or “anti-target” trials are gathered. Second, simultaneous orthogonal contrasts of neural responses may be constructed to yield a spherical visualization that exhaustively maps out data patterns while preserving the topology of interval-scale data type. (I termed it “Locus Analysis” upon the recommendation of a UM colleague who believed such naming would be good for my tenure case. The method was later applied to analyze other neuronal population data.) Third, I have developed a time-series method to separate stimulus and response waveforms in stimulus-locked and response-locked ERP averages. With the help of a diligent student (Gang Yin), this method has recently been refined to control for noise robustly as well as extended to situations with more than two behavioral time-marks.
§ Reflecting my interest in preference ordering (ranking) and its aggregation are two papers in JMP in which 1) I re-invented the well-known combinatoric object “permutahedron” (after its first discovery by Schulz in 1911!), but this time gave the proper interpretation of the space of Borda counts and linked it with other choice paradigms; 2) we re-examined Chichilnilsky’s topological approach to Arrow’s impossibility theorem and, invoking non-Hausdorff topology (due primarily to my student Matt Jones), proved possibility results when null-preference is included in the output (but not input) of social aggregation.
§ I collaborated with a highly motivated student Greg Stevens (with whom I lost contact now unfortunately) in proposing a dynamic system model of infant attachment characterizing the dyadic interaction between the infant and primary care-giver based on an arousal and a soothing neural chemical system. The insights were his and the math was mine.
§ Game theory as normative theory of social interaction is both fascinating and disappointing in illuminating the notion of rationality. In joint work with my student Matt Jones, we showed how cooperation in a prisoners dilemma game can arise as an individually-optimal strategy if players (with non-zero decision horizon) consider possible future interactions in maximizing own reward. With respect to recursive (“I think you think I think …” type or theory-of-mind) reasoning in games, my students (Trey Hedden and others) and I developed a series of three-step, sequential move games that allow us to test shallow vs deep (recursive) reasoning in subjects. Aside from laboratory studies, I also invoked the notion of meta-games for modeling military strategic engagements (in collaboration with UM student Alex Chavez).
§ My interests in reinforcement learning started around 1997 (owing to stimulating discussions with an exchange student Min Chang). Some discoveries were made, but “perished” before they became “published”, due to the rapid pace of the field around that time and some personal distraction. The one finding that did not get sufficient attention by the field, but recently published, is the observation that adaptive learning by individual agents via Law of Effect (underlying reinforcement learning) is, on the ensemble level, equivalent to naïve Bayesian dynamics (put alternatively, replica dynamics has a Bayesian interpretation). In recent collaborations with Kent Berridge and Wayne Aldridge, we looked at coding of ventral pallidal neurons and examined the role of motivation (incentive salience) in RL.
§ With the help of an incredibly talented mathematics postdoc, Dr. Haizhang Zhang (who got his from Yuesheng Xu of Syracuse), who basically single-handedly completed all technical proofs in our joint papers, we scrutinized the role of inner product operator played in reproducing-kernel Hilbert space (RKHS) method, and found that the entire reproducing-kernel framework can be generalized to a Banach space B. The trick is to invoke the notion of semi-inner-product on B, which is essentially the pull-back of the linear functional (in B*) via a generally non-linear duality mapping between B and B*. The notion of s.i.p. turns out to be useful for defining and studying frames and Riesz basis in Banach space. The notion of s.i.p can be further generalized to that of a generalized s.i.p. reflecting a generalized duality mapping. Etc, etc. This is an exciting area that we are continuously working on.
§ Information geometry provides an elegant framework for understanding asymmetric “distance” (called “divergence function”) with many applications in statistics, engineering, optimization, and machine learning. In work (that I am proud in providing), I show how a general class of divergence functions (that encompass most known families) can be constructed from a strictly convex function, and that such convex-based divergence functions precisely induces the alpha-geometry of Amari et al. I then further elucidated the connection of convex functions and the dual differential geometry they induce. I also extended the information geometric construction, in terms of formulae, of metric and affine connections to infinite-dimensional manifold (admittedly, with non-trivial caveats related to topology of infinite-dimensional spaces).
§ By carefully separating reference-duality and representational duality in the alpha-geometry, I advance the notion of reference-representation biduality that I believe is of fundamental importance to information science. This work is in parallel to my strong urge for a deeper appreciation of duality in functional analysis and kernel methods (via semi-inner-product and duality mapping).