One of the most important problems in visual perception is that of visual invariance: how are objects perceived to be the same despite undergoing transformations such as translations, rotations or scaling? In this paper, we describe a Bayesian method for learning invariances based on Lie group theory. We show that previous approaches based on first-order Taylor series expansions of inputs can be regarded as special cases of the Lie group approach, the latter being capable of handling in principle arbitrarily large transfonnations. Using a matrixexponential based generative model of images, we derive an unsupervised algorithm for learning Lie group operators from input data containing infinitesimal transfonnations.

One of the most important problems in visual perception is that of visual invariance: how are objects perceived to be the same despite undergoing transformations such as translations, rotations or scaling? In this paper, we describe a Bayesian method for learning invariances based on Lie group theory. We show that previous approaches based on first-order Taylor series expansions of inputs can be regarded as special cases of the Lie group approach, the latter being capable of handling in principle arbitrarily large transfonnations. Using a matrixexponential based generative model of images, we derive an unsupervised algorithm for learning Lie group operators from input data containing infinitesimal transfonnations.

One of the most important problems in visual perception is that of visual invariance: howare objects perceived to be the same despite undergoing transformations such as translations, rotations or scaling? In this paper, we describe a Bayesian method for learning invariances based on Lie group theory. We show that previous approaches based on first-order Taylor series expansions of inputs can be regarded as special cases of the Lie group approach, the latter being capable ofhandling in principle arbitrarily large transfonnations. Using a matrixexponential basedgenerative model of images, we derive an unsupervised algorithm for learning Lie group operators from input data containing infinitesimal transfonnations.

To understand human object recognition, it is essential to understand how objects are represented in human visual memory. A central component in object recognition is the matching of the stored object representation with that derived from the image input.But the nature of the object representation has to be inferred from recognition performance, by taking into account the contribution from the image information. When evaluating human performance, how can one separate the con- 830 ZLiu and D. Kersten tributions to performance of the image information from the representation? Ideal observer analysis provides a precise computational tool to answer this question. An ideal observer's recognition performance is restricted only by the available image information and is otherwise optimal, in the sense of statistical decision theory, irrespective of how the model is implemented.

Further, the greater the similarity between objects, the stronger is the dependence on object appearance, and the more important twodimensional (2D) image information becomes. These findings, however, do not rule out the use of 3D structural information in recognition, and the degree to which 3D information is used in visual memory is an important issue. Liu, Knill, & Kersten (1995) showed that any model that is restricted to rotations in the image plane of independent 2D templates could not account for human performance in discriminating novel object views. We now present results from models of generalized radial basis functions (GRBF), 2D nearest neighbor matching that allows 2D affine transformations, and a Bayesian statistical estimator that integrates over all possible 2D affine transformations. The performance of the human observers relative to each of the models is better for the novel views than for the familiar template views, suggesting that humans generalize better to novel views from template views. The Bayesian estimator yields the optimal performance with 2D affine transformations and independent 2D templates. Therefore, models of 2D affine matching operations with independent 2D templates are unlikely to account for human recognition performance.

Biophysical modeling studies have previously shown that cortical pyramidal cells driven by strong NMDA-type synaptic currents and/or containing dendritic voltage-dependent Ca or Na channels, respond more strongly when synapses are activated in several spatially clustered groups of optimal size-in comparison to the same number of synapses activated diffusely about the dendritic arbor [8]- The nonlinear intradendritic interactions giving rise to this "cluster sensitivity" property are akin to a layer of virtual nonlinear "hidden units" in the dendrites, with implications for the cellular basis of learning and memory [7, 6], and for certain classes of nonlinear sensory processing [8]- In the present study, we show that a single neuron, with access only to excitatory inputs from unoriented ONand OFFcenter cells in the LGN, exhibits the principal nonlinear response properties of a "complex" cell in primary visual cortex, namely orientation tuning coupled with translation invariance and contrast insensitivity_ We conjecture that this type of intradendritic processing could explain how complex cell responses can persist in the absence of oriented simple cell input [13]- 84 B. W. Mel, D. L. Ruderman and K. A. Archie

Biophysical modeling studies have previously shown that cortical pyramidal cells driven by strong NMDA-type synaptic currents and/or containing dendritic voltage-dependent Ca or Na channels, respondmore strongly when synapses are activated in several spatially clustered groups of optimal size-in comparison to the same number of synapses activated diffusely about the dendritic arbor [8]- The nonlinear intradendritic interactions giving rise to this "cluster sensitivity" property are akin to a layer of virtual nonlinear "hiddenunits" in the dendrites, with implications for the cellular basis of learning and memory [7, 6], and for certain classes of nonlinear sensory processing [8]- In the present study, we show that a single neuron, with access only to excitatory inputs from unoriented ONand OFFcenter cells in the LGN, exhibits the principal nonlinear response properties of a "complex" cell in primary visual cortex, namely orientation tuning coupled with translation invariance andcontrast insensitivity_ We conjecture that this type of intradendritic processing could explain how complex cell responses can persist in the absence of oriented simple cell input [13]- 84 B. W. Mel, D. L. Ruderman and K. A. Archie

Biophysical modeling studies have previously shown that cortical pyramidal cells driven by strong NMDA-type synaptic currents and/or containing dendritic voltage-dependent Ca or Na channels, respond more strongly when synapses are activated in several spatially clustered groups of optimal size-in comparison to the same number of synapses activated diffusely about the dendritic arbor [8]- The nonlinear intradendritic interactions giving rise to this "cluster sensitivity" property are akin to a layer of virtual nonlinear "hidden units" in the dendrites, with implications for the cellular basis of learning and memory [7, 6], and for certain classes of nonlinear sensory processing [8]- In the present study, we show that a single neuron, with access only to excitatory inputs from unoriented ONand OFFcenter cells in the LGN, exhibits the principal nonlinear response properties of a "complex" cell in primary visual cortex, namely orientation tuning coupled with translation invariance and contrast insensitivity_ We conjecture that this type of intradendritic processing could explain how complex cell responses can persist in the absence of oriented simple cell input [13]- 84 B. W. Mel, D. L. Ruderman and K. A. Archie

Coherent oscillatory activity in large networks of biological or artificial neuralunits may be a useful mechanism for coding information pertaining to a single perceptual object or for detailing regularities within a data set. We consider the dynamics of a large array of simple coupled oscillators under a variety of connection schemes. Of particular interest is the rapid and robust phase-locking that results from a "sparse" scheme where each oscillator is strongly coupled to a tiny, randomly selected, subset of its neighbors.

One ofthe major theoretical issues in neural computation is to understand how this efficiency is reached given the constraints imposed by the biological hardware. Part of the problem [2] is simply to give an informative representation ofthe visual world using a limited number of neurons, each of which has a limited information capacity. The information capacity of the visual system is determined in part by the spatial transfer characteristics, or "receptive fields," of the individual cells. From a theoretical point of view we can ask if there exists an optimal choice for these receptive fields, a choice which maximizes the information transfer through the system given the hardware constraints. We show that this optimization problem has a simple formulation which allows us to use the intuition developed through the variational approach to quantum mechanics. In general our approach leads to receptive fields which are quite unlike those observed forcells in the visual cortex. In particular orientation selectivity is not a generic prediction. The optimal filters, however, depend on the statistical properties ofthe images we are trying to sample. Natural images have a symmetry - scale invariance [4] - which saves the theory: The optimal receptive fields for sampling of natural images are indeed orientation selective and bear a striking resemblance to observed receptive field characteristics in the mammalian visual cortex as well as the retinal ganglion of lower vertebrates.