Optimality of the Johnson-Lindenstrauss Lemma

Larsen, Kasper Green; Nelson, Jelani

doi:10.1109/focs.2017.64

Cited by 60 publications

(46 citation statements)

References 17 publications

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“…By [56,78], the vertices of (n − 1)-simplex embed isometrically into any infinite dimensional Banach space, so we have thus justified the bound (16), and hence in particular the first lower bound on k α n ( 2 ) in (10). As we already explained, the second lower bound (for the almost-isometric regime) on k α n ( 2 ) in (10) is due to the very recent work [148]. The upper bound on k α n ( 2 ) in (10), namely that in (25) we can take…”

Section: Finite Subsets Of Hilbert Spacementioning

confidence: 87%

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Metric Dimension Reduction: A Snapshot of the Ribe Program

NAOR

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Section: Finite Subsets Of Hilbert Spacementioning

confidence: 87%

“…The present article is focused on embeddings that permit large errors, and in particular in ways to prove impossibility results even if large errors are allowed. For this reason, we will not describe here the ideas of the proof in [148] that pertains to the almost-isometric regime.…”

Section: 22mentioning

confidence: 99%

Metric Dimension Reduction: A Snapshot of the Ribe Program

NAOR

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…The MLP training finds a distance-preserving lowdimensional embedding function f specific to a given x t . The existence of such embedding is guaranteed by the Johnson-Lindenstrauss lemma [13], [14], under a more general setting which does not have the restriction about the embedding being specific to a given vector.…”

Section: A Learning the Del Function With Multi-layer Perceptronmentioning

confidence: 99%

DaiMoN: A Decentralized Artificial Intelligence Model Network

Teerapittayanon

Kung

2019

2019 IEEE International Conference on Blockchain (Blockchain)

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We introduce DaiMoN, a decentralized artificial intelligence model network, which incentivizes peer collaboration in improving the accuracy of machine learning models for a given classification problem. It is an autonomous network where peers may submit models with improved accuracy and other peers may verify the accuracy improvement. The system maintains an append-only decentralized ledger to keep the log of critical information, including who has trained the model and improved its accuracy, when it has been improved, by how much it has improved, and where to find the newly updated model. DaiMoN rewards these contributing peers with cryptographic tokens. A main feature of DaiMoN is that it allows peers to verify the accuracy improvement of submitted models without knowing the test labels. This is an essential component in order to mitigate intentional model overfitting by model-improving peers. To enable this model accuracy evaluation with hidden test labels, DaiMoN uses a novel learnable Distance Embedding for Labels (DEL) function proposed in this paper. Specific to each test dataset, DEL scrambles the test label vector by embedding it in a low-dimension space while approximately preserving the distance between the dataset's test label vector and a label vector inferred by the classifier. It therefore allows proof-of-improvement (PoI) by peers without providing them access to true test labels. We provide analysis and empirical evidence that under DEL, peers can accurately assess model accuracy. We also argue that it is hard to invert the embedding function and thus, DEL is resilient against attacks aiming to recover test labels in order to cheat. Our prototype implementation of DaiMoN is available at https://github.com/steerapi/daimon.

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“…The first one is to improve the sketching dimension. This, however, is known to be impossible for various regimes, see [12,63,65,72] and very recently for any embedding method by Larsen and Nelson [73]. The other direction is to make the sketching matrix sparse.…”

Section: Lemma 11 (Distributional Johnson-lindenstrauss Lemma) There mentioning

confidence: 99%