Learning Artistic Signatures: Symmetry Discovery and Style Transfer

TL;DR

This work tackles the lack of a principled definition for artistic style by proposing a dual framework: style arises from local textures described by Gram matrices $G^l$ and global structure encoded by symmetry via Lie algebra generators $\mathfrak{g}$. It introduces a tractable combined metric $D = (1-\lambda) d_{\text{texture}}(G,\bar{G}) + \lambda d_{\text{global}}(\mathfrak{g},\mathfrak{g}')$ and validates it on a dataset of 50 artists and 1,700 paintings, using methods that compute Lie generators with LieGG, distances on Grassmann manifolds, and Gram-based texture distances from a VGG-19 backbone. Hierarchical clustering, bootstrapping, and Mantel tests demonstrate that incorporating symmetry information improves alignment with art-historical groupings and yields more robust, interpretable distinctions between movements and artists. The results offer a theoretically grounded pathway for more interpretable style transfer and a framework for analyzing and distinguishing stylistic nuances across art history. This work thus bridges perceptual notions of style with a mathematically grounded representation that can inform future generative modeling and art-analytic tools.

Abstract

Despite nearly a decade of literature on style transfer, there is no undisputed definition of artistic style. State-of-the-art models produce impressive results but are difficult to interpret since, without a coherent definition of style, the problem of style transfer is inherently ill-posed. Early work framed style-transfer as an optimization problem but treated style as a measure only of texture. This led to artifacts in the outputs of early models where content features from the style image sometimes bled into the output image. Conversely, more recent work with diffusion models offers compelling empirical results but provides little theoretical grounding. To address these issues, we propose an alternative definition of artistic style. We suggest that style should be thought of as a set of global symmetries that dictate the arrangement of local textures. We validate this perspective empirically by learning the symmetries of a large dataset of paintings and showing that symmetries are predictive of the artistic movement to which each painting belongs. Finally, we show that by considering both local and global features, using both Lie generators and traditional measures of texture, we can quantitatively capture the stylistic similarity between artists better than with either set of features alone. This approach not only aligns well with art historians' consensus but also offers a robust framework for distinguishing nuanced stylistic differences, allowing for a more interpretable, theoretically grounded approach to style transfer.

Figures (6)

• Figure 1: Mondrian's Composition ii (left), Rembrandt's The Anatomy Lesson (middle), Warhol's Campbell's Soup Cans (right)
• Figure 2: Learned symmetry transformations from different artistic movements. The symmetries were learned on Impressionist (top) and Renaissance (bottom) paintings.
• Figure 3: Phylogenic trees of artistic movements. Hierarchical clustering of artists using (a) only Gram matrices and (b) both Gram matrices and Lie generators. We see that incorporating symmetry information in (b) yields a higher proportion of neighboring artists from the same art-historical movement (highlighted).
• Figure 4: Timeline of Art Movements:
• Figure 5: Artistic Movements after Impressionism, in roughly chronological order from top to bottom.

Theorems & Definitions (7)

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