Faculty Bibliography

Efficient coding provides a powerful principle for explaining early sensory coding. Most attempts to test this principle have been limited to linear, noiseless models, and when applied to natural images, have yielded oriented filters consistent with responses in primary visual cortex. Here we show that an efficient coding model that incorporates biologically realistic ingredients - input and output noise, nonlinear response functions, and a metabolic cost on the firing rate - predicts receptive fields and response nonlinearities similar to those observed in the retina. Specifically, we develop numerical methods for simultaneously learning the linear filters and response nonlinearities of a population of model neurons, so as to maximize information transmission subject to metabolic costs. When applied to an ensemble of natural images, the method yields filters that are center-surround and nonlinearities that are rectifying. The filters are organized into two populations, with On- and Off-centers, which independently tile the visual space. As observed in the primate retina, the Off-center neurons are more numerous and have filters with smaller spatial extent. In the absence of noise, our method reduces to a generalized version of independent components analysis, with an adapted nonlinear "contrast" function; in this case, the optimal filters are localized and oriented.

We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a constrained convex optimization problem, in which the full set of dictionary coefficients represents a linear approximation of the signal, the auxiliary coefficients are constrained so as to only represent translated features, and sparsity is imposed on the primary coefficients using an L1 penalty. The basis pursuit denoising (BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus refer to our methodology as continuous basis pursuit (CBP). We develop two implementations of CBP for a one-dimensional translation-invariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods, demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which in turn offers substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.

Rainstorms, insect swarms, and galloping horses produce 'sound textures'--the collective result of many similar acoustic events. Sound textures are distinguished by temporal homogeneity, suggesting they could be recognized with time-averaged statistics. To test this hypothesis, we processed real-world textures with an auditory model containing filters tuned for sound frequencies and their modulations, and measured statistics of the resulting decomposition. We then assessed the realism and recognizability of novel sounds synthesized to have matching statistics. Statistics of individual frequency channels, capturing spectral power and sparsity, generally failed to produce compelling synthetic textures; however, combining them with correlations between channels produced identifiable and natural-sounding textures. Synthesis quality declined if statistics were computed from biologically implausible auditory models. The results suggest that sound texture perception is mediated by relatively simple statistics of early auditory representations, presumably computed by downstream neural populations. The synthesis methodology offers a powerful tool for their further investigation

The human capacity to recognize complex visual patterns emerges in a sequence of brain areas known as the ventral stream, beginning with primary visual cortex (V1). We developed a population model for mid-ventral processing, in which nonlinear combinations of V1 responses are averaged in receptive fields that grow with eccentricity. To test the model, we generated novel forms of visual metamers, stimuli that differ physically but look the same. We developed a behavioral protocol that uses metameric stimuli to estimate the receptive field sizes in which the model features are represented. Because receptive field sizes change along the ventral stream, our behavioral results can identify the visual area corresponding to the representation. Measurements in human observers implicate visual area V2, providing a new functional account of neurons in this area. The model also explains deficits of peripheral vision known as crowding, and provides a quantitative framework for assessing the capabilities and limitations of everyday vision

Humans are good at performing visual tasks, but experimental measurements have revealed substantial biases in the perception of basic visual attributes. An appealing hypothesis is that these biases arise through a process of statistical inference, in which information from noisy measurements is fused with a probabilistic model of the environment. However, such inference is optimal only if the observer's internal model matches the environment. We found this to be the case. We measured performance in an orientation-estimation task and found that orientation judgments were more accurate at cardinal (horizontal and vertical) orientations. Judgments made under conditions of uncertainty were strongly biased toward cardinal orientations. We estimated observers' internal models for orientation and found that they matched the local orientation distribution measured in photographs. In addition, we determined how a neural population could embed probabilistic information responsible for such biases

The local spatiotemporal pattern of light on the retina is often consistent with a single translational velocity which may also be interpreted as a superposition of spatial patterns translating with different velocities. Human perception reflects such interpretations, as can be demonstrated using stimuli constructed from a superposition of two drifting gratings. Depending on a variety of parameters, these stimuli may be perceived as a coherently moving plaid pattern or as two transparent gratings moving in different directions. Here, we propose a quantitative model that explains how and why such interpretations are selected. An observer's percept corresponds to the most probable interpretation of noisy measurements of local image motion, based on separate prior beliefs about the speed and singularity of visual motion. This model accounts for human perceptual interpretations across a broad range of angles and speeds. With optimized parameters, its components are consistent with previous results in motion perception

Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with known statistics (i.e., a likelihood function) and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, which expresses the optimal least squares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein's unbiased risk estimator (SURE), which provides such an expression for the additive gaussian noise case. This error expression may then be optimized over noisy measurement samples, in the absence of supervised training data, yielding a generalized SURE-optimized parametric least squares (SURE2PLS) estimator. In the special case of a linear parameterization (i.e., a sum of nonlinear kernel functions), the objective function is quadratic, and we derive an incremental form for learning this estimator from data. We also show that combining the NEBLS form with its corresponding generalized SURE expression produces a generalization of the score-matching procedure for parametric density estimation. Finally, we have implemented several examples of such estimators, and we show that their performance is comparable to their optimal Bayesian or supervised regression counterparts for moderate to large amounts of data.

Consider the decomposition of a signal into features that undergo transformations drawn from a continuous family. Current methods discretely sample the transformations and apply sparse recovery methods to the resulting finite dictionary. These methods do not exploit the underlying continuous structure, thereby limiting the ability to produce sparse solutions. Instead, we employ interpolation functions which linearly approximate the manifold of scaled and transformed features. Coefficients are interpreted as interpolation weights, and we formulate a convex optimization problem for obtaining them, enforcing both reconstruction accuracy and sparsity. We compare our method, which we call continuous basis pursuit (CBP) with the standard basis pursuit approach on a sparse deconvolution task. CBP yields substantially sparser solutions without sacrificing accuracy, and does so with a smaller dictionary. We conclude that for signals generated by transformation-invariant processes, a representation that explicitly accommodates the transformation(s) can yield sparser and more interpretable decompositions.