On the Intrinsic Dimensionality of Image Representations (1803.09672v2)

Abstract: This paper addresses the following questions pertaining to the intrinsic dimensionality of any given image representation: (i) estimate its intrinsic dimensionality, (ii) develop a deep neural network based non-linear mapping, dubbed DeepMDS, that transforms the ambient representation to the minimal intrinsic space, and (iii) validate the veracity of the mapping through image matching in the intrinsic space. Experiments on benchmark image datasets (LFW, IJB-C and ImageNet-100) reveal that the intrinsic dimensionality of deep neural network representations is significantly lower than the dimensionality of the ambient features. For instance, SphereFace's 512-dim face representation and ResNet's 512-dim image representation have an intrinsic dimensionality of 16 and 19 respectively. Further, the DeepMDS mapping is able to obtain a representation of significantly lower dimensionality while maintaining discriminative ability to a large extent, 59.75% TAR @ 0.1% FAR in 16-dim vs 71.26% TAR in 512-dim on IJB-C and a Top-1 accuracy of 77.0% at 19-dim vs 83.4% at 512-dim on ImageNet-100.

On the Intrinsic Dimensionality of Image Representations

The paper "On the Intrinsic Dimensionality of Image Representations" provides a rigorous exploration of the intrinsic dimensionality (ID) of deep neural network (DNN) based image representations. The authors endeavor to address two pivotal questions within this domain: determining the intrinsic dimensionality of a given image representation and discovering a non-linear mapping to translate the ambient representation into its minimal intrinsic space, without significantly compromising its discriminative power.

Core Contributions:

The paper makes several noteworthy contributions to the field of computer vision and image representation:

1. Intrinsic Dimensionality Estimation:
   • The paper marks the first attempt to quantitatively determine the intrinsic dimensionality of image representations derived from DNNs. Intrinsic dimensionality is defined as the minimum number of parameters required to encapsulate the entirety of information present in a representation.
   • By adopting a topological dimensionality estimation technique based on the geodesic distance, the paper proposes that these deep representations, though typically high-dimensional, possess a significantly lower intrinsic dimensionality. For example, it was found that SphereFace's 512-dimension face representation and ResNet's 512-dimension image representation have intrinsic dimensions of 16 and 19, respectively.
2. DeepMDS: A Deep Neural Network Based Approach:
   • The authors introduce DeepMDS, an unsupervised deep learning framework rooted in multidimensional scaling (MDS), which transforms the high-dimensional ambient representation to a compact intrinsic space.
   • DeepMDS is shown to effectively reduce dimensionality while largely preserving the discriminative potential of the original features. For instance, the DeepMDS mapping achieved a 59.75% True Accept Rate (TAR) at a 0.1% False Accept Rate (FAR) in a 16-dimensional space, compared to 71.26% at 512 dimensions on the IJB-C dataset.

Analytical Approach:

The research hinges on leveraging a topological perspective for estimating ID, utilizing geodesic distances—which are better suited to capture the manifold structure of data compared to Euclidean distances—as key metrics. Moreover, the paper distinguishes between linear dimensionality, traditionally estimated through PCA, and intrinsic dimensionality, which considers the manifold's actual complexity within the high-dimensional space.

One challenge addressed is verifying that a given mapping accurately reflects the intrinsic dimensionality while preserving the essential structure for classification tasks. The paper systematically demonstrates that the DeepMDS mapping achieves this goal, outperforming traditional methods such as PCA and Isomap in maintaining the classification performance.

Implications and Future Directions:

The implications of this research are multifold:

• From a theoretical standpoint, the findings offer a substantial reevaluation of how dimensionality is perceived in deep learning contexts. By realigning focus towards intrinsic dimensionality, future research can explore more efficient and compact models, potentially reshaping the development of neural network architectures themselves.
• Practically, understanding the intrinsic properties of image representations facilitates optimal resource allocation—crucial for memory and computational efficiency in large-scale deployments of AI systems, such as in real-time image retrieval or face verification tasks.

The paper beckons further investigation into the mechanisms by which intrinsic dimensionality influences generalization capabilities and how algorithms might be directly crafted to exploit these compact embeddings. The work also poses intriguing possibilities for extending this intrinsic dimensionality framework to other domains and modalities beyond image data.

In summation, this paper presents a foundational step toward a nuanced comprehension of deep representation spaces, emphasizing the necessity of moving towards inherently compact and effective representations.