Composable Core-sets for Determinant Maximization: A Simple Near-Optimal Algorithm
Pith reviewed 2026-05-25 01:16 UTC · model grok-4.3
The pith
A local search algorithm constructs composable core-sets for determinant maximization with an O(k)^{2k} approximation guarantee.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Composable core-sets for determinant maximization can be built by a local search algorithm that achieves an approximation bound of O(k)^{2k}. This bound is nearly optimal and improves on the O(C^{k^2}) guarantee proved for the greedy algorithm in the same composable setting. The construction supports the MAP inference task for determinantal point processes and works in massive data models where data are partitioned across machines.
What carries the argument
Local search procedure adapted to produce composable core-sets
If this is right
- Determinant maximization instances can be solved approximately in one pass over partitioned data using only small core-sets from each machine.
- The same local-search construction supplies a practical alternative when the optimal O(k)^k bound is too expensive to achieve.
- Greedy remains usable but carries a provably weaker guarantee that grows double-exponentially with k.
- Implementation on standard data sets shows the local-search core-sets retain high quality while keeping running time modest.
Where Pith is reading between the lines
- The approach may extend to other submodular diversity objectives that admit local-search approximations.
- In distributed fairness applications the core-sets could reduce communication while preserving diversity guarantees.
- Runtime comparisons on larger synthetic partitions would test whether the O(k)^{2k} bound appears in practice or remains loose.
Load-bearing premise
The local search steps can be modified for composable core-sets without introducing extra approximation factors that grow with the number of machines or the input distribution.
What would settle it
A concrete data partition across machines where the composed local-search core-set returns a determinant value worse than O(k)^{2k} times the global optimum.
read the original abstract
``Composable core-sets'' are an efficient framework for solving optimization problems in massive data models. In this work, we consider efficient construction of composable core-sets for the determinant maximization problem. This can also be cast as the MAP inference task for determinantal point processes, that have recently gained a lot of interest for modeling diversity and fairness. The problem was recently studied in [IMOR'18], where they designed composable core-sets with the optimal approximation bound of $\tilde O(k)^k$. On the other hand, the more practical Greedy algorithm has been previously used in similar contexts. In this work, first we provide a theoretical approximation guarantee of $O(C^{k^2})$ for the Greedy algorithm in the context of composable core-sets; Further, we propose to use a Local Search based algorithm that while being still practical, achieves a nearly optimal approximation bound of $O(k)^{2k}$; Finally, we implement all three algorithms and show the effectiveness of our proposed algorithm on standard data sets.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies composable core-sets for determinant maximization (equivalently, MAP inference in DPPs). It gives an O(C^{k^2}) guarantee for the greedy algorithm in the composable setting, proposes a local-search procedure claimed to achieve O(k)^{2k} approximation while remaining practical, and reports experiments on standard data sets. The O(k)^{2k} bound is presented as nearly optimal relative to the Õ(k)^k result of IMOR'18.
Significance. If the local-search analysis is correct, the work supplies a simple, implementable method whose theoretical guarantee is only a k-factor worse than the prior optimum while retaining practicality; the experimental section supplies concrete evidence of effectiveness on real data.
major comments (1)
- [Local-search composability proof] The composability analysis (likely §4 or the proof of Theorem 3) must explicitly bound any multiplicative factor introduced by the merge step across m partitions. The abstract and weakest assumption do not rule out an extra factor polynomial in m or dependent on spectral properties of the input; without this, the claimed O(k)^{2k} guarantee is not yet shown to be independent of the distributed setting.
minor comments (2)
- [Greedy analysis] Clarify the constant C in the greedy bound O(C^{k^2}) and its dependence on input parameters.
- [Experiments] Add a short comparison table of the three algorithms' approximation ratios, running times, and core-set sizes.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the constructive comment on the composability analysis. We address the point below and will revise the manuscript accordingly.
read point-by-point responses
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Referee: [Local-search composability proof] The composability analysis (likely §4 or the proof of Theorem 3) must explicitly bound any multiplicative factor introduced by the merge step across m partitions. The abstract and weakest assumption do not rule out an extra factor polynomial in m or dependent on spectral properties of the input; without this, the claimed O(k)^{2k} guarantee is not yet shown to be independent of the distributed setting.
Authors: We agree that the proof of Theorem 3 should contain an explicit statement bounding the contribution of the merge step. In the current analysis the local-search procedure is first run independently on each of the m partitions (producing core-sets of size k) and then run once more on their union; the O(k)^{2k} guarantee follows from the standard local-search analysis applied to this final merged instance. Because the local-search ratio depends only on the cardinality k and not on the ambient dimension, the number of partitions m, or any spectral quantity of the original matrix, the merge step introduces no additional multiplicative factor. Nevertheless, to make this independence fully transparent we will add a short paragraph immediately after the statement of Theorem 3 that isolates the merge step and confirms the ratio remains O(k)^{2k} independently of m. revision: yes
Circularity Check
No circularity: bounds derived independently from prior work
full rationale
The paper presents new approximation analyses for the Greedy algorithm (O(C^{k^2})) and a Local Search variant (O(k)^{2k}) for composable core-sets on determinant maximization. These are stated as results of the current work, distinct from the cited IMOR'18 Õ(k)^k bound. No equations reduce a claimed prediction to a fitted input by construction, no self-citation forms the load-bearing justification for the central claims, and no ansatz or uniqueness theorem is imported from the authors' prior work. The derivation chain is self-contained against external benchmarks.
discussion (0)
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