Learning to ask a new question

When I began working on manifold fitting, a recurring question was not whether the method would work, but whether it counted as statistics at all. To many, statistical work meant estimating parameters within a given model, while estimating geometry itself belonged elsewhere. Yet statistics has long reached beyond parameter estimation. A question that looks natural from one discipline can look unfamiliar from another, and before making progress I first had to argue that it was worth asking.

That experience shifted my priorities. I grew less concerned with finding the right disciplinary label and more concerned with what would go unexplained if the question were left unasked. In a recent commentary in The Straits Times, I argued that recovering geometric structure could help us interrogate medical AI. Here I want to describe how that question took shape, because how research questions form matters as much as how they are answered.

When geometry becomes the question

In analysing complex data, geometry usually plays a supporting role: it helps us organise observations, reduce dimensions or separate groups. The question that came to drive my work inverts that role. What if geometric structure is itself something we should recover from the data — geometry as signal rather than scaffolding? Manifold fitting, which seeks to recover a lower-dimensional structure from noisy observations in high-dimensional space, gives this idea a concrete statistical form.

This changes what counts as an answer. A method can classify observations accurately without recovering the structure we set out to understand. So the test is no longer only whether a method works, but whether the task captures what we actually want to know.

Before a project has a name

This conviction shapes how I work with students. On Saturdays, those who wish to join meet with me for open-ended conversations about problems we find important. We ask why a problem deserves attention and whether any part of it is ready to be explored. An idea need not arrive with a method or a clear route to a solution, and I share in that uncertainty. Some problems lie beyond what I currently know how to tackle yet still seem worth thinking about. My own experience has made me wary of forcing a familiar formulation too early.

Applications keep these conversations grounded. An observation a method cannot explain may expose a weakness in the method or reveal something important about the phenomenon, sometimes changing the question itself. Many ideas never become projects; some lose their appeal after further reading, others stay out of reach. But there is a quieter form of progress worth valuing: when a vague concern sharpens into a question precise enough to show where to begin. The aim is for students to find questions they have their own reasons to pursue, including ones that lead beyond where I first pointed.

Staying with a question

Talk of using AI to generate research questions tends to skip the harder part. Finding a question is one thing and deciding it is worth staying with is another. That judgment rarely arrives all at once. It develops through reading, discussion and attempts that expose what one has misunderstood. A question can remain worth pursuing even when no route to an answer is visible and, equally, familiarity can supply good reasons to let it go.

When near-term results looked unlikely, I was sometimes advised to move on. It was genuinely hard to tell whether slow progress signalled a flawed direction or simply how much I still had to understand; the absence of quick results settled nothing. As I prepare to speak on “Geometry as Signal” in Germany this winter, I keep returning to the question that started this work, and to how much my understanding of it has changed.

That is what I bring to our Saturday conversations. Where an unfamiliar question will lead is often impossible to see in advance. Learning not to dismiss it too quickly, and learning when it is worth staying with, is a part of research I am still navigating myself.

Yao Zhigang is an Associate Professor in the Department of Statistics and Data Science at the National University of Singapore’s Faculty of Science, with a joint appointment in the Department of Mathematics