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A Sample-Based Approach for Hierarchical Information-Theoretic Compression of Probabilistic Occupancy Grids
Zhenyu Jin, Daniel T. Larsson
TL;DR
Large probabilistic occupancy grids need principled compression that respects computational constraints, while optimal Q-tree search can be computationally expensive. This paper uses an anytime, MCTS-inspired sample-based search to construct hierarchical abstractions, retaining approximately 88% of Q-tree's information with about 11% of its computation time in a real-world compression experiment.
Problem
Information-optimal Q-tree compression is computationally expensive for large-scale probabilistic occupancy grids used in robotics.
Source: §I · pp. 1–2Method
Results
Approximately 88% of the information preserved by Q-tree search was retained using about 11% of its computation time in the real-world compression experiment.
Source: B. Application to Real-World Occupancy… · p. 6Takeaways & Limitations
Abstract
from arXiv · showhide
We develop a sample-based framework for constructing information-driven hierarchical multi-resolution representations of probabilistic occupancy grids. Recent methods compute information-optimal abstractions via dynamic-programming-based exhaustive recursions, which become computationally prohibitive for large-scale grids and are ill-suited to robotics applications. To address this limitation, we introduce a sample-based strategy inspired by Monte Carlo Tree Search (MCTS) that incrementally constructs hierarchical abstractions through statistical estimation rather than exhaustive enumeration. The proposed method is anytime in nature, allowing computation to be terminated at any stage to produce a valid compressed representation. We compare our approach with the information-optimal Q-tree search algorithm and demonstrate its effectiveness in rapidly generating abstractions of large real-world probabilistic occupancy grids.
I. INTRODUCTION
Information-theoretic methods make occupancy-grid compression principled, but optimal Q-tree search is computationally expensive on large grids. This paper proposes an anytime sample-based alternative for constructing hierarchical abstractions.
I. INTRODUCTION
- Heuristic compression can reduce representation size and resource burdens, but offers no formal guarantees on compressed size or preserved occupancy information.Source: §I · p. 1
- The information bottleneck frames hierarchical grid design as balancing predictive information retention against compression through mutual information.Hierarchical grids can be treated as structured deterministic encoders, enabling information-theoretic evaluation of retained occupancy information.Source: §I · p. 1
- Q-tree search returns an optimal information-bottleneck compression but is computationally expensive on large real-world grids.Its recursive bottom-up traversal evaluates the full hierarchy.Source: §I · pp. 1–2
- The proposed MCTS-inspired sample-based method incrementally constructs abstractions, can return a valid representation at any stage, and avoids exhaustive bottom-up recursion.It supports a trade-off between computational effort and abstraction quality.Source: §I · p. 2
II. PROBLEM FORMULATION & KEY CHALLENGES
The paper casts occupancy-grid compression as information-bottleneck optimization over feasible hierarchical trees. It motivates a sample-based alternative because Q-tree search requires full-tree evaluation and cannot naturally return intermediate solutions.
II. PROBLEM FORMULATION & KEY CHALLENGES
- The encoder is restricted to feasible hierarchical trees, with finest-resolution cells as source outcomes and occupancy as the relevance variable.Each tree deterministically maps source cells to leaf nodes.Source: §II · p. 2
- The information-bottleneck objective seeks hierarchical representations that retain occupancy information while compressing the grid, with source, relevance, and compressed variables linked through mutual information.Source: §II · p. 2
- Q-tree search obtains an optimal solution through bottom-up recursion, requiring generation and evaluation of the entire finest-resolution tree before returning an abstraction.Source: §II · p. 2
- The proposed MCTS-inspired approach estimates action values through sampled rollouts, adaptively focuses computation on high-utility regions, and can return a valid solution at an intermediate stage.Limiting the rollout count controls computational effort.Source: §II · p. 3
III. A SAMPLE-BASED ALGORITHM FOR THE DESIGN OF HIERARCHICAL ABSTRACTIONS
The algorithm turns hierarchical compression into incremental node selection by exploiting the additive structure of the information-bottleneck objective.
III. A SAMPLE-BASED ALGORITHM FOR THE DESIGN OF HIERARCHICAL ABSTRACTIONS
- The information-bottleneck objective decomposes over local partition refinements, so refining a region can be assigned a node-centric action value.Source: §III · p. 3
- The algorithm incrementally selects which nodes to expand, framing abstraction construction as sequential decision-making.Source: §III · p. 3
- A state–action pair represents refining a specific region of the hierarchical tree.Source: §III · p. 3
A. Value-Based Formulation of the Tree-Search Problem
The compression problem is cast as tree search: states encode current leaves, actions refine a leaf or stop, and rewards track objective change. The objective's node-level structure supports a bottom-up value recursion for an optimal solution.
A. Value-Based Formulation of the Tree-Search Problem
- Each state is the current tree's leaf set, and an action expands one leaf or selects −1 to stop.Expanding a leaf replaces it with its children, producing the next tree state.Source: A. Value-Based Formulation of the… · p. 3
- The one-step reward is the change in Jα between trees differing by one leaf expansion: ΔJα(a) = ΔIY(a) − αΔIX(a).The information increments use the Jensen–Shannon divergence among child distributions and the Shannon entropy of the merger distribution.Source: A. Value-Based Formulation of the… · p. 3
- Although the tree-state space is large and expansion sequences can reach the same state, disjoint leaf regions and node-local expansion costs permit a node-level reformulation.This structure makes the node-based value function used by Q-tree search possible.Source: A. Value-Based Formulation of the… · pp. 3–4
- Q-tree search computes an optimal compression by bottom-up recursion, comparing pruning a node with expanding it and adding its children's future values.The resulting node values guide top-down expansion until no node offers positive value.Source: A. Value-Based Formulation of the… · p. 4
B. Evaluation and Back-propagation of Sample-Based Rollouts
The method estimates node values from finite rollout samples rather than exhaustively evaluating the tree. It combines guided forward exploration with reward-based updates and backpropagation to construct an abstraction incrementally.
B. Evaluation and Back-propagation of Sample-Based Rollouts
- The algorithm uses Monte Carlo sampling to estimate node value rapidly while avoiding evaluation of nodes unlikely to remain in the compressed tree.Combining tree-state UCB and node-level ε-greedy exploration focuses rollouts on informative regions.Source: B. Evaluation and Back-propagation of… · p. 4
- Because node values suffice to determine state-action values, the algorithm samples directly in node space using forward rollout evaluation followed by backward value updates.The procedure assumes a finite rollout budget M.Source: B. Evaluation and Back-propagation of… · p. 4
- A two-step exploration strategy uses UCB to choose a promising leaf and ε-greedy selection to continue through its children or terminate the rollout.The UCB rule balances exploration of under-visited regions with sample allocation toward promising leaves.Source: B. Evaluation and Back-propagation of… · p. 4
- Each rollout’s one-step rewards update the starting node’s value estimate, which is then backpropagated to its ancestors.The estimate uses the node’s visit count and iteration number; sampling then resumes using updated action values.Source: B. Evaluation and Back-propagation of… · p. 4
C. A Monte-Carlo Algorithm for Abstraction Design
The algorithm incrementally builds a Monte Carlo tree to guide sampling toward high-value regions, then uses the resulting node estimates to construct the compression tree. Its comparison figure depicts Q-tree and sample-based abstractions for the same example grid.
C. A Monte-Carlo Algorithm for Abstraction Design
- The method initializes a Monte Carlo tree at the root and adds discovered states while estimating their action values through rollout and backpropagation.α controls the information–compression trade-off, while ε and κ regulate exploration.Source: C. A Monte-Carlo Algorithm for… · p. 5
- After the rollout budget is exhausted, the algorithm expands nodes with positive estimated values from the root until no further positive-value expansions remain.The procedure uses UCB to explore stored states and evaluates newly encountered states with rollouts.Source: C. A Monte-Carlo Algorithm for… · p. 5
- Figure 1 compares a 128 × 128 example environment, its Q-tree abstraction, and a sample-based abstraction produced with M = 3000 at α = 0.0067.Source: Figure 1 · p. 5
IV. RESULTS AND DISCUSSION
The evaluation compares the sample-based approach with optimal Q-tree search on a sample grid and examines regional multi-resolution compression on real-world LiDAR grids. The reported sample-grid abstractions closely match Q-tree search, and the approach generates regional compressions from LiDAR data.
IV. RESULTS AND DISCUSSION
- The sample-based method produces sample-grid abstractions that closely match those obtained by exhaustive Q-tree search.Source: §IV · p. 5
- The approach demonstrates the ability to generate regional multi-resolution compressions from probabilistic grids constructed using real-world LiDAR data.Source: §IV · p. 5
- The evaluation covers two scenarios: comparison with optimal Q-tree search on a sample grid and regional compression of real-world LiDAR grids.Source: §IV · p. 5
A. Comparison with Q-tree Search
The sample-based method approaches Q-tree search as rollout budgets increase, and at M = 5000 its leaf counts and retained predictive information closely match the optimal solutions. The comparison varies α to examine reward, error, and the compression–information trade-off.
A. Comparison with Q-tree Search
- The comparison uses a 128 × 128 grid, assumes a uniform source distribution, and varies α with ε = 0.2 and κ = 1.Grayscale intensity represents each cell’s occupancy probability.Source: A. Comparison with Q-tree Search · p. 5
- At M = 5000, the sample-based method produces leaf counts highly consistent with Q-tree search and abstractions retaining nearly identical predictive information.These similarities extend to the multi-resolution representations produced by both methods.Source: A. Comparison with Q-tree Search · pp. 5–6
- As rollout budgets increase, sample-based solutions become nearly identical to Q-tree search and recover its information–complexity frontier with sufficient samples.Fig. 2 reports reward, error relative to Q-tree search, and normalized information-plane curves across α and rollout budgets.Source: A. Comparison with Q-tree Search · p. 5, Figure 2 · p. 6
B. Application to Real-World Occupancy Grid Compression
On real-world occupancy-grid regions, the sample-based method trades some retained information for substantially lower computation time than Q-tree search. The evaluation uses LiDAR-derived data and an early-stopping heuristic during rollouts.
B. Application to Real-World Occupancy Grid Compression
- The evaluation uses a 2048 × 2048 LiDAR map from a ground robot, selects navigation-relevant regions RA, RB, and RC, and applies early rollout stopping when a sampled subtree has positive value.Fig. 4 shows example sample-based compressions for RA and RB.Source: B. Application to Real-World Occupancy… · p. 6, Figure 4 · p. 6
- The method can be terminated at any point during its M rollouts to return a valid representation, whereas interrupted Q-tree search cannot produce a valid solution.Early termination may yield a representation with less retained information.Source: B. Application to Real-World Occupancy… · p. 6
- About 11% of Q-tree search computation time yields approximately 88% of its retained information, a roughly nine-fold speedup with about 12% less information.The comparison covers three regions and uses α1 = 0.0005, α2 = 0.004, α3 = 0.0067, and M = 5500; the reported gains use no hardware acceleration.Source: B. Application to Real-World Occupancy… · p. 6, Figure 5 · p. 7
V. CONCLUSION
The conclusion presents a sample-based, MCTS-inspired approach for information-theoretic compression of probabilistic occupancy grids. It reports anytime operation and rapid abstraction generation on large-scale real-world grids.
V. CONCLUSION
- The approach builds hierarchical multi-resolution occupancy-grid compressions incrementally using sampled value estimates inspired by Monte Carlo Tree Search.Source: §V · p. 7
- Computation can be stopped early, directing the available sampling budget toward regions that appear more promising while still producing a valid representation.Source: §V · p. 7
- Comparison with information-optimal Q-tree search demonstrates rapid generation of abstractions for large-scale probabilistic occupancy grids derived from real-world data.Source: §V · p. 7