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The Cauchy-Schwarz divergence for Poisson point processes
Hung Gia Hoang, Ba-Ngu Vo, Ba-Tuong Vo, Ronald Mahler
TL;DR
Point-process similarity measures can be difficult to compute analytically, often requiring expensive approximations. The paper extends Cauchy-Schwarz divergence to Poisson point processes and derives related distance identities. It establishes that the divergence is half the squared L2 distance between intensities and relates Bhattacharyya distance to Hellinger distance.
Problem
Except for trivial special cases, similarity measures for point processes often cannot be computed analytically and require expensive approximations such as Monte Carlo.
Source: §I · p. 1Method
Results
For Poisson point processes, Cauchy-Schwarz divergence is half the squared L2 distance between intensities, while Bhattacharyya distance equals squared Hellinger distance between intensity measures.
Source: §V · p. 8Takeaways & Limitations
The identities provide analytic similarity measures for Poisson models, and the paper illustrates their use in multi-target sensor management.
Source: §I · p. 1, §IV · p. 6
Abstract
from arXiv · showhide
In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared $\mathbf{L^{2}}$-distance between their intensity functions. Extension of this result to mixtures of Poisson point processes and, in the case where the intensity functions are Gaussian mixtures, closed form expressions for the Cauchy-Schwarz divergence are presented. Our result also implies that the Bhattachryaa distance between the probability distributions of two Poisson point processes is equal to the square of the Hellinger distance between their intensity measures. We illustrate the result via a sensor management application where the system states are modeled as point processes.
I. INTRODUCTION
Similarity measures for point processes have been difficult to compute analytically, often requiring expensive approximations. The paper develops analytic results for Poisson point processes and relates their similarity measures to intensity functions.
I. INTRODUCTION
- Except for trivial special cases, point-process similarity measures often cannot be computed analytically and require expensive approximations such as Monte Carlo.Source: §I · p. 1
- The paper relates Cauchy-Schwarz divergence, Bhattacharyya distance, and Hellinger distance for Poisson point processes, extending the divergence result to mixtures and Gaussian-mixture intensities.It also illustrates the divergence in a multi-target tracking sensor-management application involving the PHD filter.Source: §I · p. 1
A. Point processes
- A point process is a random variable taking values in finite subsets of the state space; for a Poisson process, counts in disjoint regions are independent and have means determined by its intensity.The paper restricts its treatment to simple-finite point processes, regarded as random finite sets.Source: A. Point processes · p. 2
B. The Cauchy-Schwarz divergence
The Cauchy-Schwarz divergence measures the difference between probability densities through their angle in an L2 space. It is symmetric, nonnegative, and connected to other divergence measures and applications.
B. The Cauchy-Schwarz divergence
- The Cauchy-Schwarz divergence is defined for two probability densities using their inner products and the logarithm of a normalized cross-inner-product.Source: B. The Cauchy-Schwarz divergence · p. 2
- The normalized inner product is the cosine of the angle between the densities in L2, and the divergence is symmetric, nonnegative, and zero when the densities are equal.Source: B. The Cauchy-Schwarz divergence · p. 2
- For Gaussian mixtures, Cauchy-Schwarz divergence can be evaluated in closed form, motivating its use in information-theoretic applications such as machine learning and pattern recognition.Source: B. The Cauchy-Schwarz divergence · p. 2
III. THE CAUCHY-SCHWARZ DIVERGENCE FOR POISSON
The paper defines a unit-consistent Cauchy-Schwarz divergence for point processes and derives its relationship to the distance between Poisson-process intensity functions. The result also gives a geometric interpretation and quantifies how intensity error relates to distributional difference.
A. Cauchy-Schwarz divergence for Poisson point processes
- A naive set-integral inner product of FISST densities is invalid because its cardinality terms have incompatible units; the divergence instead uses probability densities relative to a unitless reference measure.Source: A. Cauchy-Schwarz divergence for… · p. 3
- For two Poisson point processes with L2 intensity functions, the Cauchy-Schwarz divergence is half the squared distance between their intensities.The proposition specifies intensities measured in units of K^-1.Source: A. Cauchy-Schwarz divergence for… · p. 3
- The squared distance between intensity functions is unit-invariant when the state-space measurement unit changes.The factor K compensates for the corresponding change in intensity values.Source: A. Cauchy-Schwarz divergence for… · pp. 3–4
- The secant of the angle between Poisson-process probability densities equals the exponential of half the squared distance between their intensity functions.Source: A. Cauchy-Schwarz divergence for… · p. 4
- The result quantifies that reducing error between intensity functions reduces the difference between the corresponding Poisson distributions.Source: A. Cauchy-Schwarz divergence for… · p. 4
B. Gaussian Mixture Intensities
For Gaussian-mixture intensities, the paper gives an analytic expression for the Cauchy-Schwarz divergence. This makes intensity-function reduction based on L2 error mathematically justified and supports approximating the divergence to arbitrary desired accuracy.
B. Gaussian Mixture Intensities
- For Poisson processes with Gaussian-mixture intensities, the Cauchy-Schwarz divergence has an analytic expression based on Gaussian-density evaluations.Source: B. Gaussian Mixture Intensities · p. 4
- The expression's computation is quadratic in the number of Gaussian components and cubic in the state dimension using standard Gauss-Jordan matrix inversion.The stated complexity assumes the larger mixture has Nv components; the optimized Coppersmith-Winograd algorithm reduces the dimension exponent to 2.373.Source: B. Gaussian Mixture Intensities · p. 4
- The result mathematically justifies Gaussian-mixture reduction based on L2 error and allows the divergence to be approximated to any desired accuracy.The passage relies on Gaussian mixtures being able to approximate any density to any desired accuracy.Source: B. Gaussian Mixture Intensities · p. 4
C. Mixture of Poisson point processes
The paper extends its Cauchy-Schwarz divergence result to mixtures of Poisson point processes. When each component has a Gaussian-mixture intensity, the divergence can be evaluated analytically.
C. Mixture of Poisson point processes
- The Cauchy-Schwarz divergence between mixtures of Poisson point processes is derived from their weighted-sum probability densities.The section introduces mixtures as weighted sums of Poisson point-process densities and states a corollary giving their divergence.Source: C. Mixture of Poisson point processes · p. 5
- When each component intensity is a Gaussian mixture, the mixture divergence can be evaluated analytically by substitution into the stated expression.The component intensities are specified in units of K^-1.Source: C. Mixture of Poisson point processes · p. 5
D. Bhattacharyya distance for Poisson point processes
The paper relates the Bhattacharyya distance between two Poisson point processes to the Hellinger distance between their intensity measures. It also gives an analytic expression for the Gaussian-intensity case.
D. Bhattacharyya distance for Poisson point processes
- The Bhattacharyya coefficient is invariant to the choice of reference measure, and the paper defines the distance for probability measures using a dominating measure.The Cauchy-Schwarz divergence, by contrast, is not necessarily invariant to the reference measure.Source: D. Bhattacharyya distance for Poisson… · p. 5
- Unlike the Cauchy-Schwarz divergence, the Bhattacharyya distance does not require square-integrable probability densities.The passage attributes this property to the square roots of probability densities always being square-integrable.Source: D. Bhattacharyya distance for Poisson… · p. 5
- The Bhattacharyya distance between two Poisson point processes equals the squared Hellinger distance between their intensity measures.The result is stated for intensity measures with densities relative to Lebesgue measure.Source: D. Bhattacharyya distance for Poisson… · pp. 5–6
- The distance can also be interpreted as the difference between the processes’ expected point counts and the Bhattacharyya coefficient of their intensity measures.Each intensity mass is the expected number of points in its respective process.Source: D. Bhattacharyya distance for Poisson… · p. 6
- For Gaussian intensities, the Bhattacharyya distance has an analytic expression based on the Gaussian Bhattacharyya coefficient.The passage introduces the result for two Poisson point processes with Gaussian intensities and specifies Gaussian intensity functions.Source: D. Bhattacharyya distance for Poisson… · p. 6
IV. APPLICATION TO MULTI-TARGET SENSOR
The paper illustrates its divergence result in multi-target sensor management, modeling system states as point processes. It presents the Cauchy-Schwarz divergence as a reward function in a linear Gaussian multi-target setting.
IV. APPLICATION TO MULTI-TARGET SENSOR
- The illustrated system assumes a linear Gaussian multi-target model, with a finite-set hidden state partially observed as another finite set.The system dynamics, detection, and false alarms are described in the appendix.Source: §IV · p. 6
- The Cauchy-Schwarz divergence is presented as a useful reward function for multi-target sensor management.The application models system states as point processes or random finite sets.Source: §IV · p. 6
- Sensor management involves recursively propagating a posterior or tractable approximation, then optimizing an objective over admissible sensor actions.The passage notes that full posterior propagation is generally intractable, while the GM-PHD filter efficiently propagates the intensity for linear Gaussian multi-target systems.Source: §IV · p. 6
A. Cauchy-Schwarz divergence based reward
The reward replaces the generally intractable R´enyi-divergence computation with a Cauchy-Schwarz reward based on the squared L2 distance between predicted and posterior intensities.
A. Cauchy-Schwarz divergence based reward
- The single-step look-ahead policy selects the control command that maximizes expected reward over possible future measurements.A computationally cheaper alternative maximizes reward for an ideal predicted measurement, assuming no clutter, perfect detection, and negligible measurement noise.Source: A. Cauchy-Schwarz divergence based… · p. 6
- The R´enyi-based approach requires integrals in infinite-dimensional spaces, which are generally intractable.The proposed computation instead uses standard integrals on the finite-dimensional state space.Source: A. Cauchy-Schwarz divergence based… · p. 7
- The Cauchy-Schwarz reward for Poisson multi-target densities is the squared L2 distance between predicted and posterior intensities.With a GM-PHD filter, the reward can be evaluated in closed form.Source: A. Cauchy-Schwarz divergence based… · p. 7
- The sensor-control policy chooses the command that maximizes the ideal reward for the predicted measurement.This policy uses the ideal-predicted-measurement approximation described for the cheaper reward calculation.Source: A. Cauchy-Schwarz divergence based… · pp. 6–7
B. Numerical example
In the multi-target tracking example, the closed-form GM-PHD implementation outperforms the approximate SMC-PHD strategies on averaged OSPA, while the two SMC-PHD strategies perform similarly.
B. Numerical example
- The comparison evaluates Cauchy-Schwarz control with both SMC approximation and a GM implementation against an existing R´enyi SMC strategy.The SMC approximation is used for both divergences to enable a fair comparison.Source: B. Numerical example · p. 7
- The typical trajectory moves toward the targets, stays nearby, then shifts toward the existing and newly born targets.This is consistent with the stated expectation that the sensor remain near targets to obtain high detection probability.Source: B. Numerical example · p. 7, Figure 4 · p. 8
- 3.21 seconds is the average run time for the Cauchy-Schwarz GM-PHD strategy, versus 10.68 seconds for its SMC-PHD implementation and 10.62 seconds for the R´enyi SMC-PHD strategy.The authors report the closed-form Cauchy-Schwarz strategy as the fastest.Source: B. Numerical example · p. 8
- The GM-PHD strategy outperforms both SMC-PHD strategies on averaged OSPA, while the SMC-PHD strategies are virtually identical.The metric uses p = 2 and c = 100m and is averaged over 200 Monte Carlo runs.Source: B. Numerical example · p. 8
- The results suggest Cauchy-Schwarz control is at least as effective as R´enyi control, with superior performance from the GM implementation.The authors attribute the GM implementation's advantage to its closed-form solution and better filtering capability.Source: B. Numerical example · p. 8
V. CONCLUSIONS
The paper relates Poisson-process Cauchy-Schwarz divergence to intensity-function distance and extends the result to mixtures. It also links Bhattacharyya distance to Hellinger distance and illustrates the result in multi-target sensor control.
V. CONCLUSIONS
- For an appropriate reference measure, Poisson-process Cauchy-Schwarz divergence is half the squared distance between intensity functions.The result extends to mixtures of Poisson point processes, with closed-form expressions for Gaussian-mixture intensity functions.Source: §V · p. 8
- In the sensor-control comparison, the GM-PHD implementation outperforms the two SMC-PHD implementations, which yield similar results.The comparison is of averaged OSPA distance across control strategies.Source: Figure 5 · p. 8
- For Poisson point processes, Bhattacharyya distance between probability distributions equals the squared Hellinger distance between intensity measures.The paper states that this is the difference between the expected number of points per process and the Bhattacharyya coefficient of the intensity measures.Source: §V · p. 8
APPENDIX A LINEAR GAUSSIAN SYSTEM MODEL
The appendices specify a linear Gaussian multi-target model with births, deaths, detections, and clutter, then describe recursive propagation of its posterior intensity using the GM-PHD filter.
APPENDIX A LINEAR GAUSSIAN SYSTEM MODEL
- Targets survive or die, while new targets arise through spontaneous births or spawning; birth and spawned-target processes have Gaussian-mixture intensities.Surviving targets form a Multi-Bernoulli point process.Source: Appendix A · pp. 8–9
- Targets evolve independently and may be detected to generate measurements; independent Poisson clutter also contributes spurious measurements.Detections corresponding to targets form a Multi-Bernoulli point process.Source: Appendix A · p. 9
APPENDIX B POSTERIOR INTENSITY PROPAGATION
- The GM-PHD filter recursively propagates the posterior intensity; Gaussian-mixture intensities remain Gaussian mixtures through prediction and update.The appendix notes a propagation technique differing from to mitigate computational issues from negative mixture weights with state-dependent detection probability.Source: Appendix B · p. 9