Kris Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, MIT, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....many features x. For example, wemaywant to estimate a person s height based on their shoe size, their hair color, their race, their eye color, etc. The typical approach to this problem is to model the entire joint density, which of course requires avery large number of samples. Recent approaches [13,11]have proposed reducing the number of samples (and modeling components) necessary by modeling the conditional density explicitly using (respectively) decision trees and mixtures of Gaussians. We propose instead to find a factorial form for the conditional density. The quantity we are trying to ....

Kris Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, MIT, 1997.

....pdf optimizes its p(yjx)over the data. EM, on the other hand, typically optimizes p(x# y) the ability to model the data as a whole. Since resources (i.e. memory, complexity) are sparse and training examples are finite, it is preferable here to directly optimize the model s conditional likelihood [21] using CEM. In other words, wewant the learning system to be good at figuring out what Mrs. Dash will do next (i.e. use x to predict y) We are not as interested in asking the system what past eventwould haveprovoked Mrs. Dash to do what she just did (i.e. use y to get x) Consider the 4 cluster ....

A.C. Popat. Conjoint probabilistic subband modeling (phd. thesis). Technical Report 461, M.I.T. Media Laboratory, 1997.

....use of the learning system. EM, on the other hand, typically optimizes p(x# y) the ability to model the data as a whole. Since resources (i.e. memory, complexity) are sparse and training examples are finite, it is preferable here to directly optimize the model s conditional likelihood [13] [19] using CEM. In other words, wewant the learning system to be good at figuring out what Mrs. Dash will do next (i.e. use x to predict y) We are not as interested in asking the system what past eventwould have provoked Mrs. Dash to do what she just did (i.e. use y to get x) Figure 10(a) and (b) ....

A.C. Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, M.I.T. Media Laboratory, 1997.

....can offer solutions with higher conditional likelihood on test data than their joint counter parts. 5 10 15 20 4 6 8 10 12 x y 10 15 20 4 6 8 10 12 x y (a) L a = 4:2 L c a = 2:4 (b) L b = 5:2 L c b = 1:8 Figure 1: Average Joint(x# y) vs. Conditional (yjx) Likelihood Visualization Popat [6] describes a simple visualization example where 4 clusters must be fit with 2 Gaussian models as in Figure 1. Here, the model in (a) has a superior joint likelihood (L a L b ) and hence a better p(x# y) solution. However, when the models are conditioned to estimate p(yjx) model (b) is superior ....

A. Popat. Conjoint probabilistic subband modeling (phd. thesis). Technical Report 461, M.I.T. Media Laboratory, 1997.

....at the highest resolution pyramid level. These different pyramids give different trade offs between spatial and frequency resolutions. In this paper, we choose to use the Gaussian pyramid for its simplicity and greater spatial localization (a detailed discussion of this issue can be found in [19]) However, other kinds of pyramids can be used instead. Neighborhood: The neighborhood can have arbitrary size and shape; the only requirement is that it contains only valid pixels. A noncausal symmetric neighborhood, for example, can be used by extending the original algorithm with two passes ....

A. C. Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, Massachusetts Institute of Technology, 1997.

....use of the learning system. EM, on the other hand, typically optimizes p(x; y) the ability to model the data as a whole. Since resources (i.e. memory, complexity) are sparse and training examples are finite, it is preferable here to directly optimize the model s conditional likelihood [13] [19] using CEM. In other words, we want the learning system to be good at figuring out what Mrs. Dash will do next (i.e. use x to predict y) We are not as interested in asking the system what past event would have provoked Mrs. Dash to do what she just did (i.e. use y to get x) Figure 10(a) and (b) ....

A.C. Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, M.I.T. Media Laboratory, 1997.

....P M n=1 p(n;x i ;y i j Theta (t Gamma1) P N i=1 P M m=1 p(m;x i ;y i j Theta (t Gamma1) P M n=1 p(n;x i ;y i j Theta (t Gamma1) log p(m;x i ;y i j Theta t ) p(m;x i ;y i j Theta (t Gamma1) 7. 4) Let us now consider the case when we are estimating a conditional density [48]. Conditioning the discrete mixture model in Equation 7.3 results in Equation 7.5. This is a conditional density with missing data m which has been re written as a joint density with missing data m over a marginal density with missing data m. We shall show that this missing data problem can not be ....

A. Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, M.I.T. Media Laboratory, 1997.

....the pdf optimizes its p(yjx) over the data. EM, on the other hand, typically optimizes p(x; y) the ability to model the data as a whole. Since resources (memory, complexity) are sparse and training examples are nite, it is preferable here to directly optimize the model s conditional likelihood [Pop97] using CEM. In other words, we want the learning system to be good at guring out what Mrs. Dash will do next (i.e. use x to predict y) We are not as interested in asking the system what past event would have provoked Mrs. Dash to do what she just did (i.e. use y to get x) Consider the ....

A.C. Popat. Conjoint probabilistic subband modeling (phd. thesis). Technical Report 461, M.I.T. Media Laboratory, 1997.

....as co occurrence matrices ) e.g. 35, 17, 29, 31, 23] Popat and Picard [50] have developed a clustering approach for representing densities of local pixel neighborhoods. They have applied this to compression, classi cation and restoration, and have demonstrated impressive synthesis results [49]. A nal distinguishing aspect of previous texture synthesis work is the algorithm used to sample from the model. If the model is Gaussian, one can simply draw samples of white Gaussian noise, and linearly transform these to achieve the desired covariance relationships. This technique may also be ....

A C (K) Popat. Conjoint probabilistic Subband Modeling. PhD thesis, Massachusetts Institute of Technology, Program in Media Arts and Sciences, Cambridge, MA, September 1997.

....with higher conditional likelihood on test data than their joint counter parts. 5 10 15 20 4 6 8 10 12 x 10 15 20 4 6 8 10 12 x (a) L a = Gamma4:2 L c a = Gamma2:4 (b) L b = Gamma5:2 L c b = Gamma1:8 Figure 1: Average Joint (x; y) vs. Conditional (yjx) Likelihood Visualization Popat [6] describes a simple visualization example where 4 clusters must be fit with 2 Gaussian models as in Figure 1. Here, the model in (a) has a superior joint likelihood (L a L b ) and hence a better p(x; y) solution. However, when the models are conditioned to estimate p(yjx) model (b) is superior ....

A. Popat. Conjoint probabilistic subband modeling (phd. thesis). Technical Report 461, M.I.T. Media Laboratory, 1997.

.... 35, 17, 29, 31, 23] Popat and Picard [50] have developed a clustering approach for representing densities of local pixel neighborhoods (including those at different scales) They have applied this to compression, classification and restoration, and have demonstrated impressive synthesis results [49]. A final distinguishing aspect of previous texture synthesis work is the algorithm used to sample from the model. If the model is Gaussian, one can simply draw samples of white Gaussian noise, and linearly transform these to achieve the desired covariance relationships. This technique may also ....

A C (K) Popat. Conjoint probabilistic Subband Modeling. PhD thesis, Massachusetts Institute of Technology, Program in Media Arts and Sciences, Cambridge, MA, September 1997.

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A. C. Popat, Conjoint Probabilistic Subband Modeling. PhD thesis, Massachusetts Institute of Technology, 1997.

....is obtained by appropriately normalizing a Gaussian mixture estimate of the joint density of the conditioning pixels and the pixel being encoded. Although normalizing a joint mixture estimated in this way generally does not result in the best conditional density estimate of comparable complexity [9], this approach is 1 Hierarchical, coarse to ne sequencing is also possible, but multiple statistical models must then be employed, and the conditioning neighborhoods required become more complex. Furthermore, preliminary results have not demonstrated a clear performance advantage in the present ....

A. C. Popat. Conjoint Probabilistic Subband Modeling. PhD thesis, Massachusetts Institute of Technology, 1997.

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