Dimension reduction and semantic interaction support image clustering by making similarity structure visible and manipulable. Existing semantic interaction methods encode users' clustering criterion (a user-interpretable semantic dimension, e.g., action, location, or mood) from direct manipulation to steer reprojection, giving users direct control over the resulting layout. Yet they typically depend on learned embeddings or a predefined criterion. In practice, users' clustering criterion often emerges gradually and becomes refined through interaction rather than being fully clear at the outset. In this work, we present CriterionSI (Criterion-guided Semantic Interaction), a method that translates incremental drag interactions into criterion-guided reprojection. CriterionSI uses large language models to infer and refine the clustering criterion from sequential user drags, while grounding semantic interpretation in human-provided feedback rather than fixed prior assumptions. CriterionSI combines the inferred criterion with local drags to guide global reprojection. The simulation-based evaluation and usage scenario demonstrate that CriterionSI can discover and refine the target criterion from sequential interactions and progressively produce criterion-aligned clustering layouts. Our code and data are available at: https://github.com/4C79/CriterionSI.

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