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With modern computation, a normally distributed posterior may be overly restric- tive for many real world events. We argue that SLAM type state estimation systems must contend with asynchronous and hard to predict events. The unpredictabil- ity of events makes it difficult to parse all sensor data perfectly, even with well- developed software procedures. As an extreme example, consider that even hu- mans, albeit at much higher sophistication, are rarely able to correctly parse events when a magic trick is performed.

Given the unpredictability and potential obscurity of real world events, we should accept it as unlikely that a perfect front-end sensor processing system will always be available, regardless of the level of engineering effort spent. There will always be another corner case.

A perfect front-end would imply that perfect data association is achieved every time among vast amounts of sensory data. For example, a robot might think a camera is re-observing a previously detected ’Exit’ sign, but in reality this could be either true or false. There is an inherent uncertainty associated with any such identification.

Much development effort has been spent on developing conservative front-end processes that only act when a high degree of certainty about true or false asso- ciations is available. Conservative front-end processes suffer both in loss of infor-

mation, such as significant loop closures, and lack transferability between applica- tions. The really hard part comes when the uncertainty of multiple associations is roughly of the same scale and the front-end process is essentially ‘splitting hairs’ on whether or not to accept or reject potential associations.

Our discussion thus far has focused on discrete true or false decisions, a con- sequence of entrenched max-product style thinking. In the sections that follow, we will show how to efficiently deal with fully continuous uncertainty with our Multi-

modal iSAMsolution.

A multi-hypothesis approach is one avenue to mitigate the effect of binary data association uncertainties, and has been widely studied in target tracking applica- tions, Reid, Fortman, Clark, et al. [35, 60, 195]. A multi-hypothesis system reduces data association uncertainties into decisions of yes or no, and at each fork in the road a separate max-product type inference procedure is started for the alternative decision. By tracking the separate likelihood uncertainty weights of each solution, such as Huang et al. [97], one could use heuristics to momentarily select a domi- nant solution among the exponentially many options.

We can algebraically demonstrate the exponential explosion in complexity as- sociated with a full multi-hypothesis approach, by again looking at the joint proba- bility distribution. Consider the product of likelihoods as shown in eq. (5.1) where two of the likelihood functions happen to be bi-modal, i.e. sum of (a, b) two equally likely normal distributions N (µ, Σ),

[ Θ | Z ] ∝ [ Z1| Θ1] [ Z2| Θ2] Y k [ Zk| Θk] Y k0 [ Θk0] = (N1,a+ N1,b) (N2,a+ N2,b) Y k [ Zk| Θk] Y k0 [ Θk0] =Y· · ·+Y· · ·+Y· · ·+Y· · ·. (5.4) These expressions show that the distributive law, which itself is an implicit con- volution, produces four major terms in the joint probability density expression. Similarly, each additional multi-modal density results in an exponential increase in the number of summands in the joint probability expression.

We should immediately stress that many of the terms, while algebraically vis- ible, have very low probability. Between all measurements, most modes collapse since the probability of the front-end actually observing those sequence of events is extremely unlikely.

Fig. 5-3 attempts to graphically illustrate the exponential complexity associ- ated with a full multi-hypothesis approach as the upper curve. In contrast, the

Full solution multi-hypothesis & nonparametric C o m pl e xi ty Number of dimensions Range of possible multi-modal solutions parametric unimodal solution

Figure 5-3: Illustration of unimodal vs. multi-hypothesis solution, with expanding space of possible multi-modal solutions.

unimodal parametric max-product solution (presented in Section 5.2.1) is a special case where each decision hypotheses is assumed correct and final, such that no alternatives are considered in parallel solutions. The lower curve illustrates the unimodal parametric case.

Fig. 5-3 also illustrates the enormous space between parametric unimodal and full blown multi-hypothesis solutions, which we will call the range space of possi-

ble multi-modal solutions. While the figure may initially seem trivial, the author believes there is a deeper observation; where we allow the front-end process to defer a reasonable amount of association uncertainties to the back-end inference procedure and let consensus happen there.

The question becomes, what do deferred association solutions look like? For example, consider a point on the lower right just above the unimodal paramet- ric curve: (i) What is the complexity of tracking a limited number of hypothe- ses across the entire solution (FastSLAM [81]) versus tracking more modes in re- gions of the solution? (ii) Can we do better than assumed ”outlier” approaches where measurements are de-weighted as null-hypotheses, such as switch type vari-

ables[76,206,219]? (iii) Can we select a piece of the problem where clear uncertain- ties exist and then treat just that portion in some special manner, leaving the rest as a conventional max-product parametric solution? (iv) Does a common algorithm exist where we can choose the granularity of the solution based on computational resources?

The questions boil down to a deeper question on the relation between approx- imated beliefs (nonparametrics) and multi-modality. We use structure within the problem to convert a highly complex multi-hypothesis, nonparametric system into

a more tractable multi-modal solution. The Bayes tree significantly reduces com- putation while still checking all uncertainty, as captured in the factor graph, and extracting consensus among all data.

The Bayes tree allows us to solve the complete problem while working at a per clique level and considering only local interactions, which is a direct consequence of the nontrivial factorization of the factor graph. Certain inference related op- erations are more efficient on the tree structure than a naive approach operating randomly over all variables in the factor graph. For example, updating all vari- ables with a Hidden Markov Model type approach, such as a Kalman or particle filter, would terribly inefficient – as the square root smoothing and mapping algo- rithm [41, 42] showed over and above the inefficiencies of the earlier EKF-SLAM algorithm [11, 48].

We believe the correct avenue is to pursue an algorithm which can estimate multi-modal, non-Gaussian marginal posterior density functions, while maintain- ing asymptotic correctness properties. Granularity can then be varied based on available computational resources. The direct benefit of a varialbe granularity (res- olution) algorithm is the ability to relax requirements on the front-end process. Given more powerful measurement models, as in Chapter 3, the user can be more optimistic when introducing measurement information. We stress that changing to non-Gaussian posteriors does not preclude use of the Bayes (Junction) tree [113], but instead actually casts the Bayes tree as a general framework for reducing com- putational complexity, regardless of the form and shape of likelihood functions used to assemble the factor graph.

Our purpose is therefore to reevaluate the set of operating assumptions from what has previously been used, such as in Section 5.2.1, to enable dedicated solu- tions that explore the range space of possible multi-modal solutions.

The next section describes the exact interclique operations on the Bayes tree, followed by a discussion on the asymptotically correct intraclique operations in Section 5.4.