Batch Normalization: Accelerating Deep Network Training b y Reducing Internal Covariate Shift. Ioffe, Szegedy. Arxiv 2015

  1. A problem with training ANNs is that as training occurs, the distribution of inputs for higher layers changes (called covariate shift).  Here they do normalization <whitening at each layer? yes> of inputs for each mini batch.
  2. Trains faster, trains to better results, is itself a form of regularization so removes need for dropout in some cases
  3. Saturation occurs frequently as a result of covariate shift.  If we can avoid that then it may make it easier to train with larger learning rates
  4. “Batch Normalization also has a beneficial effect on the gradient flow through the network, by reducing the dependence of gradients on the scale of the parameters or of their initial values. This allows us to use much higher learning rates without the risk of divergence. Furthermore, batch normalization regularizes the model and reduces the need for Dropout (Srivastava et al., 2014). Finally, Batch Normalization makes it possible to use saturating nonlinearities by preventing the network from getting stuck in the saturated modes.”
  5. Normalization parameters must be computed inside the gradient descent step (so in batch mode, and not online).  This can be shown both theoretically and in practice
  6. Normalization is done by each input independently (this is to save computational costs, and also because there needs to be some computation that isn’t differentiable <but needs to be?>)
    1. <I guess so, later on:>  “Thus, BN transform is a differentiable transformation that introduces normalized activations into the network. “
  7. In order to make sure the normalization doesn’t ruin expressability of the layer, a constraint is that “the transformation inserted in the network can represent the identity transform”
  8. “In traditional deep networks, too-high learning rate may result in the gradients that explode or vanish, as well as getting stuck in poor local minima.”
  9. Naturally it also helps deal with scaling issues in the inputs
  10. “Moreover, larger weights lead to smaller gradients, and Batch Normalization will stabilize the parameter growth.”
  11. Because it acts as regularization, can remove the need for dropout and ReLUs, as well as other forms of regularization (such as L2 weight regularization), can also allow for slower weight decay
  12. Get state of the art results on imagenet, and reaches human-level performance
  13. “Batch Normalization adds only two extra parameters per activation, and in doing so preserves the representation ability of the network.”
  14. State that this may help with training problems that are part of RNNs

Continuous Control with Deep Reinforcement Learning. Lilicrap, Hunt, Pritzel, Heess, Erez, Tasssa, Silver. Arxiv 2015

  1. Extension of deep QL to continuous actions
  2. Actor-critic, model-free
  3. Deterministic policy gradient
  4. Show the algorithm running on 20 tasks in a physics simulator
  5. A followup to deterministic policy gradient paper
  6. Uses most of the tricks from the Atari paper plus something relatively new called batch normalization
  7. Ran algorithms directly on joint angle data as well as simulated camera images
  8. Alg is called deep deterministic policy gradient
  9. They are able to use same parameters for the direct state information as well as visual data
  10. Method is simple and pretty straightforward actor-critic
  11. They compare results to a planner that has access to a generative model
  12. DDPG can sometimes outperform the planner that accesses the generative model, even in some cases when working only from the visual data
  13. DPG requires:
    1. A parameterized actor function which is a mapping from states to actions
    2. Critic, which has Q-function
  14. NFCQA is basically the same as DPG but uses an NN as a FA.  Issue is it uses batch learning which doesn’t scale well.
    1. The minibatch version of this algorithm is equivalent to the original formulation of DPG
  15. Do “soft” updates of the network which makes weights change more slowly but helps prevent divergence
    1. “This simple change moves the relatively unstable problem of learning the action-value function closer to the case of supervised learning, a problem for which robust solutions exist. “
    2. Did this both for the policy and Q
  16. Method of batch normalization is an approach that helps deal with issue that different parts of the state vector may have different scales and meanings
    1. <From what I can tell here, this looks like it basically does minibatch whitening of the data, is it really such a new idea?  Need to check the paper where it is introduced.>
  17. They just add Gaussian noise to the actor in order to do exploration
  18. Most of the problem looks like came from MuJoCo, some in 2d and some in 3d, but they also did racing in Torcs
  19. Similar to the atari papers they use the last 3 frames of data to represent state
  20. Visual data is downsampled to 64×64, 80-bit
  21. “Surprisingly, in some simpler tasks, learning policies from pixels is just as fast as learning using the low-dimensional state descriptor. This may be due to the action repeats making the problem simpler. It may also be that the convolutional layers provide an easily separable representation of state space, which is straightforward for the higher layers to learn on quickly.”
  22. The planner they compare against is iLQG which <I think> is a locally-optimal controller
    1. It needs not only the model but also its derivatives
  23. “The original DPG paper evaluated the algorithm with toy problems using tile-coding and linear function approximators. It demonstrated data efficiency advantages for off-policy DPG over bothon- and off-policy stochastic actor critic. It also solved one more challenging task in which a multijointed octopus arm had to strike a target with any part of the limb. However, that paper did not demonstrate scaling the approach to large, high-dimensional observation spaces as we have here.”
  24. “It has often been assumed that standard policy search methods such as those explored in the present work are simply too fragile to scale to difficult problems [17]. Standard policy search is thought to be difficult because it deals simultaneously with complex environmental dynamics and a complex policy. Indeed, most past work with actor-critic and policy optimization approaches have had diffi- culty scaling up to more challenging problems [18]. Typically, this is due to instability in learning wherein progress on a problem is either destroyed by subsequent learning updates, or else learning is too slow to be practical.”
  25. Similar to guided policy search <?>
  26. Looks like Q estimates are close to the returns that the policies generate

Distinguishing Conjoint and Independent Neural Tuning for Stimulus Features With fMRI Adaptation. Drucker, Kerr, Aguirre. Innovative Methodology 2009.

  1. “We describe an application of functional magnetic resonance imaging (fMRI) adaptation to distinguish between independent and conjoint neural representations of dimensions by examining the neural signal evoked by changes in one versus two stimulus dimensions and considering the metric of two-dimension additivity.”
  2. “Do different neurons represent individual stimulus dimensions or could one neuron be tuned to represent multiple dimensions?”
  3. “This study describes an application of functional magnetic resonance imaging (fMRI) to distinguish conjoint from independent representation of two stimulus dimensions within a spatially restricted population of neurons.”
  4. “If the recovery for a combined change is simply the additive combination of the recovery for each dimension in isolation, we take this as evidence for independent neural populations. When the neural recovery for a combined change is subadditive, this may reflect populations consisting of neurons that conjointly represent the two stimulus dimensions.”
  5. “These two possibilities could be distinguished directly by measuring the tuning of individual neurons. However, the signal obtained with BOLD fMRI averages the population neural response from a voxel, making this measurement unavailable.”
  6. “To distinguish conjoint and independent tuning in this case, we must measure the properties of the neural population using adaptation methods.”
  7. “In summary, we may distinguish between conjoint and independent tuning of neurons in a population by comparing the recovery from adaptation for combined transitions to that seen for isolated transitions along each stimulus dimension.”
  8. “In theory, one could conduct the test described earlier by measuring the BOLD fMRI response to three stimulus pairs: a pair that differs only in color, a pair that differs only in shape, and a pair that differs in both color and shape.”
    1. To make this robust though, more thorough sampling is needed
  9. Looking at the difference between Manhattan and Euclidian distance helps figure out how neurons respond to stimuli with multiple dimensions (additive/independent or not)
  10. Another issue that needs to be addressed before interpreting how neurons respond requires figuring out how linear changes in stimuli lead to changes in neural response — assumed to be nonlinear <I figure they measure this as they vary only 1 dimension?>
  11. They have a model to estimate the nonlinearity, <trying to deal with varying forms of nonlinearities seems complex the way they do it maybe there is a better way>
  12. “A different violation of the model assumptions occurs when the underlying neural representation is independent for the stimulus dimensions, but its neural instantiation is not aligned with the assumed dimensional axes of the study. For example, consider an experiment designed to examine the neural representation of rectangles. The stimulus space used in the experiment consists of rectangles that vary in height and width, and the experimenter models these two parameters. It may be the case, however, that a population of neurons actually has independent tuning for the sum and difference of height and width (roughly corresponding to area and aspect ratio)—a 45° rotation of the axes as modeled by the experimenter.”
  13. “In summary, when significant loading on the Euclidean contraction covariate is obtained in an experiment, an additional test is necessary to reject the possibility of independent,but misaligned, neural populations. Post hoc testing of the performance of the model under assumed rotations of the stimulus axes can distinguish between the independent, but rotated, and the conjointly tuned cases.”
  14. “Earlier, we considered how these concepts are related to receptive fields that are either linear or radially symmetric within a stimulus space. Intermediate receptive fields are possible, however, with oval shapes of varying elongation. In such cases the population would not be wholly independent, but instead represent one dimension to a greater extent than the other. These intermediate cases are considered readily within the framework of the Minkowski exponent that defines the representational space.”
  15. fmris introduce other problems related to things they are bad at measuring
  16. “The test for a conjointly tuned neural population amounts to the measurement of variance attributable to the Euclidean contraction covariate. We consider here optimizations of the approach to maximize power for this test.”
  17. Instead of sampling stimuli from a grid of parameter space, they sample from nested octagons
    1. Helps keep points at a diagonal more evenly spaced than those that are up/down which is an issue in rectangular sampling. “the dioctagonal space increases the range of the Euclidean contraction covariate, thus improving power.”
  18. “Our selection of stimuli was motivated by the psychological study of integral and separable perceptual spaces. Some visual properties of objects are apprehended separately (e.g., color and shape), whereas other dimensions are perceived as a composite (e.g., saturation and brightness); these have been termed separable and integral dimensions (Shepard 1964). We hypothesized that integral perceptual dimensions are represented by populations of neurons that represent the dimensions conjointly, whereas separable dimensions are represented by independent neural populations; similar ideas have been proposed recently”
  19. First set of experiments are on “popcorn” and “moons”
  20. Screen Shot 2015-08-05 at 12.10.34 PM
  21. There is evidence that the two dimensions used for both stimuli sets are perceptually independent
  22. <Maybe they dont actually do an experiment on the stars that vary orange to red with differing number of points and only use them as an example?  that would be a bummer>
  23. “During separate fMRI scanning sessions … Subjects were required to monitor and report the position of a bisecting line, which was randomly tilted and shifted within preset limits … to maintain attention.”
  24. “For each subject, we identified within ventral occipitotemporal cortex voxels that showed recovery from adaptation to both stimulus axes for both stimulus spaces …Most voxels were concentrated around the right posterior fusiform sulcus, corresponding to ventral LOC…”
  25. In popcorn they find the neural representation is not independent based on dimension, but for the crescents they may indeed be independent
  26. “Although a particular study may find independent tuning for a pair of stimulus dimensions, it does not automatically follow that neurons are therefore tuned “for” those axes. It remains possible that the dimensions selected for study are manifestations of some further, as yet unstudied, organizational scheme.”
  27. In discussion, mentions other features that are thought to be represented independently on a neural level
  28. “Our method amounts to using a linear model to test the metric of a space—an approach that has been considered problematic…we have argued by simulation for the validity of our model for two dimensions with 16 regularly spaced samples.”
    1. “Herein we have considered several types of nonlinearities and distortions that can exist in neural representation or recovery from adaptation. Although we find that the method is generally robust to these deviations, there naturally exists the possibility of further violations of the assumptions of the model that we have not evaluated.”
  29. “We envision the use of the metric estimation test to study the representation of stimulus properties across sensory cortical areas. By revealing the presence of independently tuned neural populations, the fundamental axes of perceptual representation might be identified. Interestingly, a given stimulus space may be represented conjointly in one region of cortex, but independently in another.”
    1. This is true of the visual system

Properties of Shape Tuning of Macaque Inferior Temporal Neurons Examined Using Rapid Serial Visual Presentation. De Baene, Premereur, Vogels. J Neurophysiology 2007.

  1. Examined macaque inferior temporal cortical neuron responses to parametrically defined shapes
  2. “we found that the large majority of neurons preferred extremes of the shape configuration, extending the results of a previous study using simpler shapes and a standard testing paradigm. A population analysis of the neuronal responses demonstrated that, in general, IT neurons can represent the similarities among the shapes at an ordinal level, extending a previous study that used a smaller number of shapes and a categorization task. However, the same analysis showed that IT neurons do not faithfully represent the physical similarities among the shapes.”
  3. Also, IT neurons adapt to stimulus distribution statistics
  4. “Single IT neurons can be strongly selective for object attributes such as shape, texture, and color, while remaining tolerant to some transformations such as object position and scale”
  5. Rapidly display images in succession without interstimulus break
  6. Other results also show that neurons seem to be tuned to activate at when shapes that come from the extremes of parameter shape are presented
  7. “Because a high number of stimuli are presented repeatedly in RSVP, this paradigm might be more sensitive to adaptive effects than classical testing paradigms in which one stimulus is presented per trial after acquisition of fixation and the intertrial interval is relatively long”
  8. <Skipping experimental details and moving on to results>
  9. <Again,> Neuron responses were tuned to extremes of the parameter space and not normally or uniformly distributed
    1. They used a number of different shape classes, and all showed this effect
  10. There was “a good overall fit between physical and neural similarities.”
  11. Although they had the issue that some dimensions were more salient than others,
  12. Screen Shot 2015-08-04 at 1.11.39 PM
  13. Did a hierarchical clustering of shapes according to neural responses and different shape classes are always together (aside from one shape class that is split in half and has another shape class “inside” it)
  14. “One issue to consider regarding the interpretation of the observed stronger responses for extreme stimuli is that the employed stimuli are likely to be suboptimal for the tested IT neurons. The critical question here is why the extreme stimuli are less suboptimal than the other stimuli given the likely high-dimensional space in which IT neurons are tuned. A satisfactory answer to this important question will require a full description of the nature of the tuning functions of IT neurons as well as knowledge about the relative position and range of the stimulus set with respect to these tuning functions. The possibility cannot be excluded that IT neurons learn the stimulus statistics of the parametric shape spaces and thus that the observed tunings depend on the stimulation history and the specific stimulus spaces. Experiment 2 demonstrated that the responses of IT neurons can indeed be modified by changes in input statistics. These effects were small in comparison to the degree of monotonic tuning, but stimulus statistics might exert a more profound effect with more extensive daily repetition of the same stimulus spaces as is the common practice in singlecell recording experiments  The MDS results clearly show that IT neurons are more sensitive for some stimulus variations (e.g., indentation; stimulus sets 3 and 4) than for others. This is in agreement with previous studies using calibrated sets of shapes…”

Representation of object similarity in human vision: psychophysics and a computational model. Cutzu, Edelman. Vision Research 1997.

  1. Visual system is robust to illumination and perspective changes.  We usually hold that we should be sensitive to changes in shape, but how do you study that in a well principled way?
  2. References to earlier work that studied 2d shape change, here considering 3d
  3. 3 main ideas about how to make pose-independent shape classification, and there are ways to test which one seems to be what we do
  4. <Mostly interested in the way they generate their shape data and properties of it, so skipping most of the other stuff>
    1.  ex/ “theories such as Shepard’s law of generalization, Nosofsky’s GCM and Ashby’s GRT”
  5. Shapes made up bodies – in all they were 70-dimensional
  6. Screen Shot 2015-07-15 at 4.58.54 PM
  7. “We remark that the nonlinearities in the image creation process led to a complicated relationship between the shape-space representation of an object and its appear- ance on the screen”

  8. “Many early studies relied on the estimation of subjective similarities between stimuli, through a process in which the ob- server had to provide a numerical rating of similarity when presented with a pair of stimuli. One drawback of this method is that many subjects do not feel comfort-

    able when forced to rate similarity on a numerical scale. Another problem is the possibility of subjects modifying their internal similarity scale as the experiment pro- gresses. We avoided these problems by employing two different methods for measuring subjective similarity: compare pairs of pairs (CPP) and delayed match to sample (DMTS).”

  9. <skipping different experimental designs, moving on to discussion>
  10. Running MDS on subject data puts points pretty much where they should be
  11. “The CPP experiments described above support the hypothesis of veridical representation of similarity, by demonstrating that it is possible to recover the true low-dimensional shape-space configuration of complex stimuli from proximity tables obtained from subjects who made forced-choice similarity judgments.”
  12. “It is important to realize that the major computa- tional accomplishment in the experiments we have de- scribed so far is that of the human visual system and not of the MDS procedure used to analyze the data.”
  13. “The detailed recovery from subject data of complex similarity patterns imposed on the stimuli supports the notion of veridical representation of similarity, dis- cussed in the introduction. Although our findings are not inconsistent with a two-stage scheme in which geometric reconstruction of individual stimuli precedes the computation of their mutual similarities, the com- putational model that accompanies these findings offers a more parsimonious account of the psychophysical data. Specifically, representing objects by their similari- ties to a number of reference shapes (as in the RBF model described in Section 6.2) allowed us to replicate the recovery of parameter-space patterns observed in human subjects, while removing the need for a prior reconstruction of the geometry of the objects.”
  14. “Assuming that perceptual simi- larities decrease monotonically with psychological space distances, multidimensional scaling algorithms derive the psychological space configuration of the stimulus points from the table of the observed similarities.”
  15. asd

Perceptual-Cognitive Explorations of a Toroidal Set of Free-Form Stimuli. Shepard, Cermak. Cognitive Psychology 1973.

<I’m just going to post images because it explains the important stuff>

Screen Shot 2015-07-15 at 3.45.37 PMScreen Shot 2015-07-15 at 3.45.27 PMScreen Shot 2015-07-15 at 3.45.46 PMScreen Shot 2015-07-15 at 3.46.11 PM

  1. But also people tended to view shapes based on what object they were most similar to (classifying them based on whether they looked like a gingerbread man, for example)
    1. “a striking aspect of the subsets is their very marked variation in size and shape in the underlying two-dimensional toroidal surface.”
    2. So these clusters don’t match the earlier contour maps either in size or shape (they are not necessarily symmetric or convex, although they seem to be L/R symmetric mostly but not up/down)
    3. Sometimes a category formed two disconnected clusters
  2. The general conclusions, here, seem to be the following: On the one hand, the underlying parameter space provides a very convenient frame- work for representing the groups into which Ss tend to sort the forms. Moreover this space is directly relevant in the sense that most of the forms sorted into any one group typically cluster together into one or two internally connected subsets in the space. But, on the other hand, the fact that the spatial representations of the spontaneously produced sub- sets vary greatly in size and shape and sometimes even consist of two or more widely separated clumps seems to establish that Experiment II taps a variety of cognitive functioning that was not operative in Experi- ment I. Just what forms will be seen as representing the same object ap- parently cannot be adequately explained solely in terms of the metric of perceptual proximity among the free forms themselves…”

  3. Each cluster can be further broken down in to subsequent subclusters
  4. Parameter space is toroidal, so top links to bottom and side to side

On correlation and budget constraints in model-based bandit optimization with application to automatic machine learning. Hoffman, Shahriari, Freitas. AISTATS 2014.

  1. Consider noisy optimization with finite samples <not yet clear if this is budget is imposed by the actor or the environment>
  2. “Bayesian approach places emphasis on detailed modelling, including the modelling of correlations among the arms. As a result, it can perform well in situations where the number of arms is much larger than the number of allowed function evaluation, whereas the frequentist counterpart is inapplicable.”
  3. “This paper draws connections between Bayesian optimization approaches and best arm identification in the bandit setting. It focuses on problems where the number of permitted function evaluations is bounded.”
  4. Applications include parameter selection for machine learning tasks
  5. “The paper also shows that one can easily obtain the same theoretical guarantees for the Bayesian approach that were previously derived in the frequentist setting [Gabillon et al., 2012].”
  6. A number of different criteria can be used in Bayesian land to select where to sample: “probability of improvement (PI), expected improvement (EI), Bayesian upper confidence bounds (UCB), and mixtures of these”
  7. Mentions work of Bubeck/Munos/Etal
  8. Tons of relevant references
  9. Also discussion in terms of simple regret
  10. But looks like they are also talking PACy
  11. Setting they consider includes GPs
  12. “As with standard Bayesian optimization with GPs, the statistics of … enable us to construct many different acquisition functions that trade-off exploration and exploitation. Thompson sampling in this setting also becomes straightforward, as we simply have to pick the maximum of the random sample from …, at one of the arms, as the next point to query.”
  13. Seems like they are really considering the finite arm case where arms have some covariance
  14. Used Bayes math to get upper and lower bounds among all arms, and then this is used to generate a bound on the simple regret
  15. “Intuitively this strategy will select either the arm minimizing our bound on the simple regret (i.e. J(t)) or the best “runner up” arm. Between these two, the arm with the highest uncertainty will be selected, i.e. the one expected to give us the most information.”
  16. The exploration parameter beta is chosen based on how often each arm is chosen and then finding something epsilon optimal
    1. Regret bound is in terms of near-optimality
  17. “Here we should note that while we are using Bayesian methodology to drive the exploration of the bandit, we are analyzing this using frequentist regret bounds. This is a common practice when analyzing the regret of Bayesian bandit methods”
  18. Can do a derivation with Hoeffding or Bernstein bounds as well (leads to analysis of case of independent arms, bounded rewards)
  19. UGap vs BayesGap – bounds are pretty much the same
  20. Have a nice empirical section where they use data from 357 traffic sensors and try to find the location with the highest speed
    1. “By looking at the results, we quickly learn that techniques that model correlation perform better than the techniques designed for best arm identification, even when they are being evaluated in a best arm identification task.”
  21. Then they use it for optimizing parameters in scikit-learn
    1. “EI, PI, and GPUCB get stuck in local minima”

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Active Model Selection. Madani, Lizotte, Greiner. UAI 2004

  1. Considers the case where there is a fixed budget
  2. Shown to be NP-Hard
  3. Consider some heuristics
  4. “We observe empirically that the simple biased-robin algorithm significantly outperforms the other algorithms in the case of identical costs and priors.”
  5. Formalize the problem in terms of coins.  You are given a set of coins with different biases, and are given a budget of number of flips to sample.  Goal is to pick the coin with the highest bias for heads.  Actually consider the case where there are priors over the distributions for each coin, so considers Bayesian case
  6. “We address the computational complexity of the problem, showing that it is in PSPACE, but also NP-hard under different coin costs.”
  7. Metric is based on regret
  8. “A strategy may be viewed as a finite, rooted, directed tree where each leaf node is a special “stop” node, and each internal node corresponds to flipping a particular coin, whose two children are also strategy trees, one for each outcome of the flip”
    1. So naturally the total number of ways this can work out is exponential
  9. “We have observed that optimal strategies for identical priors typically enjoy a similar pattern (with some exceptions): their top branch (i.e., as long as the outcomes are all heads) consists of flipping the same coin, and the bottom branch (i.e., as long as the outcomes are all tails) consists of flipping the coins in a Round-Robin fashion”
  10. Update estimates on coins according to beta distribution
  11. “The proof reduces the Knapsack Problem to a special coins problem where the coins have different costs, and discrete priors with non-zero probability at head probabilities 0 and 1 only. It shows that maximizing the profit in the Knapsack instance is equivalent to maximizing the probability of finding a perfect coin, which is shown equivalent to minimizing the regret. The reduction reveals the packing aspect of the budgeted problem. It remains open whether the problem is NP-hard when the coins have unit costs and/or uni-modal distributions”
  12. “It follows that in selecting the coin to flip, two significant properties of a coin are the magnitude of its current mean, and the spread of its density (think “variance”), that is how changeable its density is if it is queried: if a coin’s mean is too low, it can be ignored by the above result, and if its density is too peaked (imagine no uncertainty), then flipping it may yield little or no information …However, the following simple, two coin example shows that the optimal action can be to flip the coin with the lower mean and lower spread!”
  13. Even if Beta parameters of two coins are fixed, the beta parameter of a third coin make require you to choose the first or second coin depending on their values
  14. Furthermore, “The next example shows that the optimal strategy can be contingent — i.e., the optimal flip at a given stage depends on the outcomes of the previous flips.”
  15. Although the optimal algorithm is contingent, an algorithm that is not contingent may only give up a little bit on optimality
  16. Discusses a number of heuristics including biased robin and interval estimation
  17. Gittins indices are simple and optimal, but only in the infinite horizon discounted case
    1. Discusses a hack to get it to work in the budgeted case (manipulating the discount based on the remaining budget)
  18. Goes on to empirical evaluation of heuristics

Gaussian Process Dynamical Models. Wang, Fleet, Hertzmann. Nips 2006

  1. “A GPDM comprises a low-dimensional latent space with associated dynamics, and a map from the latent space to an observation space.”
  2. “We demonstrate the approach on human motion capture data in which each pose is 62-dimensional.”
  3. “we show that integrating over parameters in nonlinear dynamical systems can also be performed in closed-form. The resulting Gaussian Process Dynamical Model (GPDM) is fully defined by a set of lowdimensional representations of the training data, with both dynamics and observation mappings learned from GP regression.”
  4. As a Bayesian nonparametric, GPs make them easier to use and overfit less
  5. “Despite the large state space, the space of activity-specific human poses and motions has a much smaller intrinsic dimensionality; in our experiments with walking and golf swings, 3 dimensions often suffice.”
  6. “The Gaussian Process Dynamical Model (GPDM) comprises a mapping from a latent space to the data space, and a dynamical model in the latent space…The GPDM is obtained by marginalizing out the parameters of the two mappings, and optimizing the latent coordinates of training data.”
  7. “t should be noted that, due to the nonlinear dynamical mapping in (3), the joint distribution of the latent coordinates is not Gaussian. Moreover, while the density over the initial state may be Gaussian, it will not remain Gaussian once propagated through the dynamics.”
  8. Looks like all predictions are 1-step, can specifically set it up to use more history to make it higher-order
  9. “In effect, the GPDM models a high probability “tube” around the data.”
  10. “Here we consider a simple online method for generating a new motion, called mean-prediction, which avoids the relatively expensive Monte Carlo sampling used above.”
  11. <Wordpress ate the rest of this post.  A very relevant paper I should follow up on.>

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