Relaxing the Uniformity and Independence Assumptions Using the Concept of Fractal Dimension

Faloutsos, Christos; Kamel, Ibrahim

doi:10.1006/jcss.1997.1522

Cited by 94 publications

(130 citation statements)

References 20 publications

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“…Over the past decades, many characterizations of the ID of sets have been proposed [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Projection-based learning methods such as PCA [16] can produce as a byproduct an estimate of ID.…”

Section: B Intrinsic Dimensionalitymentioning

confidence: 99%

The vulnerability of learning to adversarial perturbation increases with intrinsic dimensionality

Amsaleg

Bailey

Barbe

et al. 2017

2017 IEEE Workshop on Information Forensics and Security (WIFS)

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Abstract-Recent research has shown that machine learning systems, including state-of-the-art deep neural networks, are vulnerable to adversarial attacks. By adding to the input object an imperceptible amount of adversarial noise, it is highly likely that the classifier can be tricked into assigning the modified object to any desired class. It has also been observed that these adversarial samples generalize well across models. A complete understanding of the nature of adversarial samples has not yet emerged. Towards this goal, we present a novel theoretical result formally linking the adversarial vulnerability of learning to the intrinsic dimensionality of the data. In particular, our investigation establishes that as the local intrinsic dimensionality (LID) increases, 1-NN classifiers become increasingly prone to being subverted. We show that in expectation, a k-nearest neighbor of a test point can be transformed into its 1-nearest neighbor by adding an amount of noise that diminishes as the LID increases. We also provide an experimental validation of the impact of LID on adversarial perturbation for both synthetic and real data, and discuss the implications of our result for general classifiers.

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Section: B Intrinsic Dimensionalitymentioning

confidence: 99%

The vulnerability of learning to adversarial perturbation increases with intrinsic dimensionality

Amsaleg

Bailey

Barbe

et al. 2017

2017 IEEE Workshop on Information Forensics and Security (WIFS)

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“…For spatial queries, the observation that the intrinsic dimensionality of a data set in many cases is lower than the representational dimensionality (due to interdependencies among attributes) is often presented as a justification of strategies for obviating the curse of dimensionality [31][32][33][34]. It should be noted that there are scenarios where correlations among attributes do exist, but the problem of discrimination of distances still applies [1].…”

Section: Problem 3: Presence Of Redundant Attributes Similarly As Witmentioning

confidence: 99%

Can Shared-Neighbor Distances Defeat the Curse of Dimensionality?

Houle

Kriegel

Kröger

et al. 2010

Lecture Notes in Computer Science

214

153

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Abstract. The performance of similarity measures for search, indexing, and data mining applications tends to degrade rapidly as the dimensionality of the data increases. The effects of the so-called 'curse of dimensionality' have been studied by researchers for data sets generated according to a single data distribution. In this paper, we study the effects of this phenomenon on different similarity measures for multiplydistributed data. In particular, we assess the performance of sharedneighbor similarity measures, which are secondary similarity measures based on the rankings of data objects induced by some primary distance measure. We find that rank-based similarity measures can result in more stable performance than their associated primary distance measures.

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“…A natural kind of such structure is clustering, but other examples include approximating the real-valued points with a low-dimensional subspace, or having low fractal dimension [4]. Given any such class of local models for the numerical data, each corresponding to an itemset in the binary data, we wish to cover the observations with itemsets, each supported by a set of rows whose numerical attributes are well explained by a local model.…”

Section: Introductionmentioning

confidence: 99%

Cross-Mining Binary and Numerical Attributes

Garriga¹,

Heikinheimo²,

Seppänen³

2007

Seventh IEEE International Conference on Data Mining (ICDM 2007)

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We consider the problem of relating itemsets mined on binary attributes of a data set to numerical attributes of the same data. An example is biogeographical data, where the numerical attributes correspond to environmental variables and the binary attributes encode the presence or absence of species in different environments. From the viewpoint of itemset mining, the task is to select a small collection of interesting itemsets using the numerical attributes; from the viewpoint of the numerical attributes, the task is to constrain the search for local patterns (e.g. clusters) using the binary attributes. We give a formal definition of the problem, discuss it theoretically, give a simple constant-factor approximation algorithm, and show by experiments on biogeographical data that the algorithm can capture interesting patterns that would not have been found using either itemset mining or clustering alone.

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Relaxing the Uniformity and Independence Assumptions Using the Concept of Fractal Dimension

Cited by 94 publications

References 20 publications

The vulnerability of learning to adversarial perturbation increases with intrinsic dimensionality

The vulnerability of learning to adversarial perturbation increases with intrinsic dimensionality

Can Shared-Neighbor Distances Defeat the Curse of Dimensionality?

Cross-Mining Binary and Numerical Attributes

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