The likelihood function of a finite mixture model is a non-convex function with multiple local maxima and commonly used iterative algorithms such as EM will converge to different solutions depending on initial conditions. In this paper we ask: is it possible to assess how far we are from the global maximum of the likelihood? Since the likelihood of a finite mixture model can grow unboundedly by centering a Gaussian on a single datapoint and shrinking the covariance, we constrain the problem by assuming that the parameters of the individual models are members of a large discrete set (e.g. estimating a mixture of two Gaussians where the means and variances of both Gaussians are members of a set of a million possible means and variances). For this setting we show that a simple upper bound on the likelihood can be computed using convex optimization and we analyze conditions under which the bound is guaranteed to be tight. This bound can then be used to assess the quality of solutions found by EM (where the final result is projected on the discrete set) or any other mixture estimation algorithm. For any dataset our method allows us to find a finite mixture model together with a dataset-specific bound on how far the likelihood of this mixture is from the global optimum of the likelihood

Although human object recognition is supposedly robust to viewpoint, much research on human perception indicates that there is a preferred or “canonical” view of objects. This phenomenon was discovered more than 30 years ago but the canonical view of only a small number of categories has been validated experimentally. Moreover, the explanation for why humans prefer the canonical view over other views remains elusive. In this paper we ask: Can we use Internet image collections to learn more about canonical views? We start by manually finding the most common view in the results returned by Internet search engines when queried with the objects used in psychophysical experiments. Our results clearly show that the most likely view in the search engine corresponds to the same view preferred by human subjects in experiments. We also present a simple method to find the most likely view in an image collection and apply it to hundreds of categories. Using the new data we have collected we present strong evidence against the two most prominent formal theories of canonical views and provide novel constraints for new theories.