Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings

arXiv:2511.15146 · stat.ML, cs.LG, math.ST, stat.TH · Submitted 2025-11-19 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Beyond Uncertainty Sets".

Tom: This paper introduces a novel framework for extending Conformal Prediction (CP) to multivariate settings by leveraging Optimal Transport (OT).

Jane: First, who's behind it and why it matters.

Title and authors: Tom: So we're diving into this paper called "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings," and it sounds like they're tackling some serious hurdles in how we measure uncertainty for complex AI outputs. It’s about taking the standard way of making prediction sets and applying it when the model gives us a whole vector of scores instead of just one number, which is a big step since most traditional conformal prediction works best with single values.

Jane: That sounds really interesting, Tom; I'm curious how they manage to handle those vector-valued scores without having to throw out the entire concept of conformal prediction. It suggests there are ways to make these new methods more robust when we deal with things like multi-output regression where you get a set of error scores instead of just one error value.

Lu: Exactly, Jane; the core problem they identify is that there's no simple, canonical way to rank vectors in R d, which is what messes up the traditional ranking mechanism CP uses for its guarantees. They propose using Optimal Transport theory to solve this by defining principled vector ranks and multivariate quantile regions that retain validity even though they are typically only known asymptotically.

Meng: From an engineering standpoint, the paper's summary focuses on how they construct these new methods, specifically building Conformal Predictive Distributions or CPDs that actually have finite-sample calibration guarantees in these multivariate tasks. That’s a major practical win because most high-dimensional models we deploy need reliable uncertainty estimates right away, not just theoretical limits.

Lalam: I see the focus on those CPDs; if we can get distributions that calibrate perfectly on a small set of data, that means our uncertainty quantification becomes much more trustworthy for real-time applications. This paper seems to be laying down a solid foundation for how we can handle complex output spaces better in the future.

Tom: Right, and what's really exciting is their proposed improvements—they tackle the tractability issue head-on by proving that the optimal assignment function is piecewise-constant across a fixed polyhedral partition of the score space. That means they're not solving a massive optimal transport problem every time they check a candidate; they can just look up where it falls in precomputed regions.

Title and authors: Jane: I like that idea about tractability; having an algorithm that moves from computationally infeasible to something fast through this geometric property is exactly what we need for deploying these complex models. It shifts the focus from solving hard optimization problems repeatedly to performing a quick lookup at test time, which is crucial for speed in production systems.

Lu: That piecewise-constant insight is key because it unlocks a tractable algorithm instead of requiring them to solve separate OT problems for every single candidate, which is what felt computationally infeasible initially. They are essentially simplifying the complexity by exploiting the structure of the solution space.

Meng: That speed is important; if we can build these prediction sets quickly, we can integrate uncertainty quantification into our live inference pipelines without slowing down response times significantly. But I wonder about the complexity they mentioned in their limitations section; they noted that it still involves solving "n + one separate n × n assignment problems," leading to a complexity of O(n four). That seems high for very large output dimensions, Meng asked.

Lalam: Wait, Tom brought up the limitation; if the complexity is O(n four), we still have a scaling issue as our output vector size gets larger. But they also showed that this boundedness allows the prediction set construction to happen in "finitely many steps" by exploiting the regularity structure of discrete transport maps. That suggests there's a way to manage the growth even with that complexity.

Tom: That’s a good point, Lalam; it sounds like they've managed to get from an intractable problem down to something computationally manageable, even if the theoretical worst-case remains high. Now, let's talk about how they build these multivariate CPDs themselves—they use a randomized map: "/Gn+one(y, τ) = (one − τ)F−n+one(S(y) + τFn+one(S(y)), where τ ∼ Uniformzero one is an auxiliary random variable."

Jane: That randomized CPD construction sounds like a clever trick to achieve the desired exact finite-sample calibration, which is something that's been hard to get in the multivariate setting. By adding that auxiliary uniform random variable τ, they ensure the resulting distribution behaves exactly like a Pointwise Independent Transform when evaluated at the true future observation Yn+one.

Title and authors: Lu: That mechanism connects nicely with other ideas; they also demonstrated that the classical one-dimensional Dempster-Hill procedure is exactly a conformal predictive system by framing it within a semi-discrete optimal transport framework. This shows a deep connection between discrete structures and continuous transport theory.

Meng: That connection to the Dempster-Hill procedure is interesting because it gives us another established mathematical structure to compare this new OT approach against, which helps ground the new methodology in existing statistical thinking. But I still need clarity on how they handle those vector ranks precisely when testing, especially given that their method relies on a "fast cell lookup at test time".

Lalam: From my perspective, the implication here is that we can build tools that provide much more granular risk assessment for complex systems because the CPD isn't just a set of points; it’s a proper distribution with guaranteed coverage. This level of probabilistic confidence really elevates what we can do with these models.

Tom: So, to wrap up this discussion on "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings," we see that the authors have successfully addressed the fundamental problem of ordering vector scores by using optimal transport, and they've developed a tractable method using polyhedral partitions for fast computation.

Jane: And they’ve shown how to construct multivariate CPDs with finite-sample calibration, which is a significant step forward because it solves a problem that was previously open in this area. It really moves us past the limitations of only being able to do this for scalar values.

Lu: The work establishes a principled way to define vector ranks and quantile regions using OT, and the randomized CPD mechanism provides an exact uniformity property when evaluated at the true future observation Yn+one. This is solid theoretical grounding for multivariate uncertainty.

Meng: While the O(n four) complexity is something we need to watch closely for deployment on massive datasets, the fact that it's bounded and computable in finitely many steps based on known regions Rk and cells Ak suggests it’s viable for many practical scenarios, provided our dimensions aren't astronomically high.

Lalam: This paper has huge implications because if we can deploy CPDs with guaranteed coverage in multivariate settings, we unlock risk-sensitive decision-making across multiple outputs in areas like autonomous systems or complex financial modeling. It provides a rigorous mathematical way to quantify uncertainty beyond simple point predictions.

Title and authors: Tom: That’s the big picture, Jane; we have a new framework that handles vector scores rigorously and efficiently using optimal transport geometry to get calibrated predictive distributions. We're really excited about how this could help us build more reliable AI systems in the real world, not just theoretical ones.

Jane: It certainly opens up new avenues for applying conformal prediction to real-world scenarios where model outputs are inherently vector-valued, like in multi-output regression tasks or ensemble predictions. We have a lot of ground to cover with this kind of mathematical rigor.

Lu: I think the future research direction here should focus on pushing that complexity bound lower or exploring how the polyhedral partition method can be adapted for even more diverse score spaces beyond just Euclidean R d, perhaps incorporating physics-aware constraints.

Meng: I agree with Lu; understanding how to make that O(n four) behavior better suited for real-world inference speed is the next big engineering hurdle we need to tackle when we start implementing this.

Lalam: This paper really shows how deep mathematical concepts like optimal transport can solve very concrete, practical problems in AI deployment, which is what I find most inspiring about this research direction.

Tom: Well, that wraps up our discussion on "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings." It’s a fascinating piece of work that provides a rigorous path toward robust uncertainty quantification in high-dimensional AI outputs.

Jane: Definitely a paper worth keeping close as we explore ways to apply these distribution-aware prediction sets across more complex machine learning architectures.

Lu: I think the connection they made between semi-discrete optimal transport and the Dempster-Hill procedure is a really neat theoretical link that could inspire future work in relating different statistical frameworks.

Meng: From an engineering viewpoint, I’m looking forward to seeing how quickly we can prototype a system using this polyhedral lookup method on our current production models.

Lalam: I'm just excited to see what other novel applications emerge from this framework; it feels like a powerful tool for building truly reliable and trustworthy AI systems that make better decisions in high-stakes environments.

The paper's summary: Tom: So, we just got the rundown on this paper, "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings," and honestly, it’s pretty dense but super interesting because it tackles how we measure uncertainty when a model spits out a whole vector of scores instead of just a single number.

Jane: It really is complex stuff, Tom; basically, the authors figured out a way to take the standard conformal prediction method and make it work for multi-output scenarios by using something called Optimal Transport theory to define proper ranking and prediction sets for those vectors.

Lu: Exactly! They address that "no unique way to order vectors" problem by using OT maps on augmented samples, which ensures the exchangeability needed for the conformal guarantee holds even in high dimensions. It’s a really creative application of geometry to probability theory.

Meng: From my side, what I’m focusing on is how they make it practical; they don't just give us theory, they build a mechanism for "multivariate CPDs with finite-sample calibration," which means the uncertainty estimates are actually trustworthy even when we only have a small amount of data to train on.

Lalam: For me, the biggest vision here is how this moves AI culture; if we can generate full predictive distributions instead of just point predictions, it lets us make much more risk-sensitive decisions in real applications, which fundamentally shifts how we trust these systems.

Tom: That’s a great way to put it, Lalam; and I think the trick they used to make the algorithm run fast is brilliant—they found that the optimal assignment function is piecewise-constant across a fixed polyhedral partition of the score space. So instead of redoing a massive calculation for every candidate, they just do a quick lookup at test time.

Jane: That part about the geometric structure allowing for a fast cell lookup sounds like it’s what makes this method actually deployable in production, Tom; it moves it from theoretical math to something that engineers can actually use quickly.

Lu: It shows how leveraging the structure of the transport problem itself can simplify things dramatically, turning an intractable problem into one that's solvable with precomputed regions. That connection between the continuous transport map and discrete partitions is really elegant.

Meng: I appreciate that tractability point; if we can reduce inference time by avoiding repeated heavy optimization, it makes integrating this kind of rigorous uncertainty quantification into live systems much more feasible for us.

Lalam: It’s exciting because it means we can move beyond just knowing *if* something is possible to knowing the *probability* mass across a whole set of possibilities, which is a huge leap in reliability for any complex AI output.

Tom: Exactly! So, we’re looking at a framework that handles vector scores rigorously and efficiently using optimal transport geometry to get calibrated predictive distributions. This opens up new avenues for applying conformal prediction to real-world scenarios where model outputs are inherently vector-valued, like in multi-output regression tasks or ensemble predictions.

Jane: And the core takeaway is that we’ve finally got a solid mathematical structure for multivariate uncertainty quantification that doesn't rely on making strong parametric assumptions about the error distribution.

Lu: The paper also showed a deep connection between semi-discrete optimal transport and classical methods like the Dempster-Hill procedure, which provides another established statistical framework to compare this new approach against.

Meng: I still have to look closely at that O(n four) complexity they mentioned; while they prove it's bounded and computable in finitely many steps, we need to ensure that for truly massive output dimensions, the practical inference speed remains acceptable for our high-throughput needs.

Lalam: This work is going to be huge because it gives us a rigorously calibrated tool for risk assessment in multi-output AI, which will definitely improve how we build and trust these systems moving forward.

The paper's improvements: Tom: So, we’ve covered how they solved the fundamental problem of ranking vector scores using optimal transport geometry, and now we’re looking at the specific improvements they propose to make this whole framework even more useful in practice.

Jane: It sounds like they are focusing on turning that complex mathematical setup into something that is genuinely fast and efficient for real-world deployment, Tom. They aren't just building a theoretical curiosity; they're suggesting concrete steps to make it usable.

Lu: I think the main improvement centers around making the algorithm tractable; they proved that by exploiting the piecewise-constant nature of that optimal assignment function, we can replace solving massive transport problems with a simple cell lookup at test time. That’s a huge win for scaling up.

Meng: From an engineering standpoint, that tractability is exactly what I need; if we can move from computationally heavy optimization to something fast and lookup-based, it makes integrating this into our current inference pipelines much more feasible without crippling response times.

Lalam: It’s exciting because this suggests we can generate risk-sensitive uncertainty sets that are not just point predictions but full predictive distributions, which means the AI can communicate its uncertainty in a way that’s directly actionable for human decision-makers.

Tom: And on the calibration side, they are pushing for methods to generate those multivariate CPDs with finite-sample calibration more robustly, which is a major hurdle they aim to clear. They want to ensure that these distributions have the correct coverage probability even when we only have a small amount of data.

Jane: That focus on finite-sample calibration is key because it solves the problem of needing huge datasets just to get reliable uncertainty estimates in multi-output tasks, which is where most current methods fall short.

Lu: They are also exploring ways to formalize the connection between this new OT approach and existing statistical ideas, like the semi-discrete optimal transport framework relating to the Dempster-Hill procedure. That helps ground this new technique in established mathematical language.

Meng: I'm interested in how they plan to handle the complexity trade-off; if the worst-case complexity is O(n four), we need to see how that performs when our output vector dimension 'n' gets quite large, something we might encounter with complex sensor fusion.

Lalam: For me, the implication is a richer AI culture where uncertainty isn't just a vague warning; it’s a quantified statement about the likelihood of different outcomes across multiple dimensions, which will lead to much more thoughtful system design.

Tom: So they are really trying to bridge that gap between having a mathematically sound method and having an algorithm that runs fast enough for production use, focusing on those structural properties of the transport maps.

Jane: It really shows a commitment to making these advanced statistical concepts practical, Tom; they’re not just proving something works on paper but providing the tools to build it.

Lu: This is interesting because the authors are also looking into future work where they might adapt this polyhedral method to handle even more complex score spaces, perhaps incorporating physical constraints into the geometric definitions.

Meng: I'll be watching those future directions closely; if they can extend this geometric approach beyond simple Euclidean R d spaces, that would open up some really interesting applications in physics-informed AI.

Lalam: This paper is pushing us toward a future where we can have high-dimensional AI systems that don't just predict an answer, but provide a complete picture of the possible outcomes with rigorous mathematical backing.

Conclusion: Tom: So, we’re wrapping up our deep dive into "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings," which really showed us how optimal transport can tackle the vector-valued score problem in conformal prediction.

Jane: It’s been fascinating watching how they managed to bridge the gap between abstract optimal transport theory and something that actually provides a practical framework for generating calibrated predictive distributions.

Lu: I think the big implication here is that it opens up a whole new area for AI research where we can move beyond scalar uncertainty into full, distribution-aware risk assessment in complex multi-output environments.

Meng: From an engineering perspective, this suggests we have a more rigorous way to define what our AI models are actually capable of saying about the range of possible outcomes simultaneously, which is vital for safety and reliability.

Lalam: For me, the vision here is that this research helps build a culture where we demand not just predictions, but mathematically guaranteed confidence levels across multiple dimensions for any output.

Tom: Exactly! The paper’s core contribution was proving that these methods can achieve exact finite-sample calibration in multivariate settings using semi-discrete optimal transport techniques.

Jane: And the way they tied it to the Dempster-Hill procedure really grounds this new math in established statistical thinking, which makes it much easier for practitioners to adopt.

Lu: It’s a very elegant mathematical structure; the Randomized CPD they built provides an exact uniformity property when evaluated at a true future observation, which is incredibly powerful for validation.

Meng: I just want to keep thinking about that O(n four) complexity they flagged; if we can figure out how to optimize that lookup process further, it changes how quickly we can iterate on these systems.

Lalam: This work will certainly inspire a new kind of uncertainty quantification in AI development, pushing us toward systems where every output has a quantified probability mass attached to it.

Tom: Absolutely! So, as we wrap up this segment on "Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings," we’ve seen how geometry can solve ordering problems and how calibration can be guaranteed.

Jane: It’s a lot of heavy theory condensed into a really solid set of practical tools for building more trustworthy AI applications.

Lu: This paper lays excellent groundwork for exploring physics-aware constraints in future work, which I think could lead to even more creative ways to define these regions beyond standard Euclidean spaces.

Meng: We’ll be paying close attention to how the polyhedral partition method scales up when we move into higher dimensions, because that's where the real engineering challenge lies for us.

Lalam: I'm just energized by this research; it shows us a path toward AI systems that don't just guess, but rigorously model their own uncertainty across every output variable.

Eugene Ndiaye

stat.ML, cs.LG, math.ST, stat.TH

Submitted: 2025-11-19

Updated: 2026-09-30

Comments: Add new algorithm with optimal complexity

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 76/100

The gist: This paper introduces a novel framework for extending Conformal Prediction (CP) to multivariate settings by leveraging Optimal Transport (OT).

Key concepts

Optimal Transport (OT): OT is used to define principled ranks for vector scores by computing a map between samples. It helps establish an exchangeability condition necessary for the conformal guarantee when dealing with multidimensional data.
Piecewise-Constant Assignment: The authors prove that the optimal assignment resulting from OT is piecewise-constant across a fixed polyhedral partition of the score space. This crucial property allows for a fast, tractable algorithm instead of solving complex OT problems repeatedly.
Multivariate CPDs: This refers to Conformal Predictive Distributions designed for multivariate tasks that have finite-sample calibration guarantees. The paper constructs these using semi-discrete optimal transport and auxiliary random variables to ensure the prediction set covers the true outcome with a guaranteed probability.
Boundedness of Prediction Set: The resulting prediction set is shown to be a finite union of convex polyhedra. This boundedness is vital because it allows the construction of the final prediction set in finitely many steps, relying on known regions and cells.

Terminology

Optimal Transport — 62× in this paper · explained in 5 episodes
Conformal Prediction — 61× in this paper · explained in 5 episodes
Calibration — 42× in this paper · explained in 7 episodes
Conformal Predictive Distribution — 33× in this paper
Exchangeability — 19× in this paper · explained in 2 episodes
Score Function — 17× in this paper · explained in 4 episodes
Probability Integral Transform — 13× in this paper
Core — 10× in this paper · explained in 2 episodes
Copula — 9× in this paper · explained in 2 episodes
Uncertainty Quantification — 9× in this paper · explained in 8 episodes
Generalization — 5× in this paper · explained in 13 episodes
Aggregation — 4× in this paper · explained in 1 episode
Alignment — 3× in this paper · explained in 1 episode
Bayesian Framework — 3× in this paper · explained in 3 episodes
Optimal Transport (OT) — 3× in this paper · explained in 4 episodes
Benchmark — 2× in this paper · explained in 2 episodes
Active Regions — 1× in this paper · explained in 1 episode
Adaptivity — 1× in this paper · explained in 1 episode
Bayesian Inference — 1× in this paper · explained in 5 episodes
Curse of Dimensionality — 1× in this paper · explained in 6 episodes
Grounding — 1× in this paper · explained in 3 episodes
Influence Functions — 1× in this paper · explained in 2 episodes
Model-Agnostic — 1× in this paper · explained in 2 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes
Quantifying Uncertainty — 1× in this paper · explained in 5 episodes
Regularization — 1× in this paper · explained in 3 episodes
Robustness — 1× in this paper · explained in 5 episodes
Theoretical Guarantees — 1× in this paper · explained in 2 episodes
Time Series Forecasting — 1× in this paper · explained in 3 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

This paper introduces a novel framework for extending Conformal Prediction (CP) to multivariate settings by leveraging Optimal Transport (OT). It addresses two fundamental limitations of standard CP: constructing valid prediction sets for vector-valued scores and building Conformal Predictive Distributions (CPDs) that possess finite-sample calibration guarantees in multivariate tasks. By utilizing OT to define principled vector ranks and quantile regions, the authors provide a method to achieve these goals distribution-free, ensuring that uncertainty quantification remains rigorous even when scores are multidimensional.

Conformal Prediction with Vector-Valued Scores

The core challenge in extending CP is handling vector-valued scores, as there is no unique or canonical way to order vectors in R d. The authors propose a solution by defining a candidate’s rank via an optimal transport map computed on an augmented sample that includes the candidate itself. This procedure ensures that the exchangeability required for the conformal guarantee is fully maintained for test points.

The key steps in this approach are:

Defining the rank of a candidate via an optimal transport map computed on a set augmented with that candidate's score.
Proving that the resulting optimal assignment is piecewise-constant across a fixed polyhedral partition of the score space, which allows for a tractable algorithm instead of solving OT problems per candidate.

Tractable Algorithm via Polyhedral Partitions

The initial approach to extending CP involves defining a continuum of OT problems (one for each candidate), which appears computationally infeasible. The authors resolve this intractability by proving that the optimal assignment function is piecewise-constant across a fixed polyhedral partition of the score space. This insight allows for a tractable algorithm that pre-computes this partition once, reducing the problem to a fast cell lookup at test time.

Multivariate Conformal Predictive Distributions (CPDs)

The paper constructs multivariate CPDs with finite-sample calibration, which is an open problem in the multivariate setting. This is achieved by defining a randomized CPD based on semi-discrete optimal transport. The randomized map, denoted as:

/Gn+1(y, τ) = (1 − τ)F−n+1(S(y) + τFn+1(S(y)), where τ ∼ Uniform[0, 1] is an auxiliary random variable. This randomized CPD is shown to be a PIT when evaluated at the true future observation Yn+1, ensuring exact finite-sample calibration. The resulting prediction set has guaranteed marginal coverage: P(Yn+1 ∈ Γ1−α(xn+1)) = 1 − α.

Optimal Transport View of Dempster-Hill Procedure

The authors demonstrate that the classical one-dimensional Dempster-Hill procedure is exactly a conformal predictive system. This connection is formalized through a semi-discrete optimal transport framework where discrete gaps in the score space are treated as Laguerre cells in the target space. The randomized map, defined as:

/T˜n+1(Zi, τ) ∼ Uσ⋆(i) = U(· Aσ⋆(i)), T˜n+1(Zi, τi) ∼ Uσ⋆(i), i ≤ n, with τ i.i.d. auxiliary randomness independent of the data,

yields exact uniformization of the predictive transform: T˜ Zn+1 (Zn+1, τ) ∼ U.

Boundedness and Tractability

The final construction leads to a semi-discrete quantile region, denoted as:

/omega sd r = Z ∈ R d: T˜Z n+1(Z, τ) ≤ r a.s. for all τ.

This set is shown to be a finite union of convex polyhedra, making it easy to compute once the regions Rk and cells Ak (or inclusion certificates Ak ⊆ B(0, r)) are known. This boundedness is crucial, as it allows the prediction set construction to be performed in finitely many steps by exploiting the regularity structure of discrete transport maps.

Connection to Hypothesis Testing

The conformal prediction set is viewed as the collection of hypotheses y that are not rejected when testing H0: Yn+1 = y vs H1: Yn+1 ≠ y. The resulting CPD arranges these individual p-values into a proper cumulative distribution function, ensuring it is monotonically increasing and that evaluating it at the true Yn+1 results in a uniformly distributed random variable. This structure requires the conformity score to satisfy specific monotonicity and limit conditions to ensure validity.

Limitations and Future Work

The tractability of the algorithm involves solving n + 1 separate n × n assignment problems, leading to a complexity of O(n 4).

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that can be made to AI systems by leveraging its core theoretical contributions:

) Improvements for AI Systems based on Optimal Transport-based Conformal Prediction (OT-CP):

Construct and utilize a multivariate predictive distribution (CPD) that is calibrated at the finite sample level.
Implement a Conformal Prediction with Vector-Valued Score framework to handle model outputs like residuals from multi-output regression or ensemble predictions, preserving exact, distribution-free coverage guarantees without resorting to dimensionality reduction (like norms).
Develop a tractable algorithm for characterizing the entire prediction set by exploiting the piecewise-constant nature of the optimal assignment function across a polyhedral partition of the score space. This allows for fast lookup at test time instead of recomputing transport maps for every candidate, making high-dimensional prediction sets computationally feasible.
Generate risk-sensitive uncertainty sets and predictive distributions rather than just point-based prediction sets by leveraging the exact uniformity property of the randomized CPD, which provides a granular likelihood for each outcome.
Apply a multivariate generalization of the Dempster-Hill procedure to provide a formal framework for generating continuous predictive distributions over high-dimensional output spaces, effectively moving beyond classical one-dimensional uncertainty quantification methods.
Utilize the semi-discrete optimal transport (SDOT) framework to generate exact multivariate probability integral transforms (PITs) by mapping the discrete empirical source measure to a continuous target distribution (like the spherical uniform law), ensuring that any derived uncertainty region has guaranteed coverage even when scores are vector-valued.

) What the Improved AI System Can Do:

The improved system will be capable of performing sophisticated, rigorously calibrated uncertainty quantification in complex, multi-output machine learning scenarios. Specifically:

Perform high-dimensional regression tasks where the output is a vector (e.g., predicting multiple sensor readings or multiple economic indicators simultaneously). The system can generate a prediction set for each output dimension that is guaranteed to contain the true outcome with specified confidence, without needing to assume any specific distribution shape for the errors.
Quantify uncertainty in ensemble models by concatenating individual model residuals into a vector score and using the OT-CP framework to define robust uncertainty regions that respect the geometry of these combined prediction errors.
Move beyond simple plausibility sets to generate full, calibrated predictive distributions for multivariate outputs. This allows decision-makers to not only know which outcomes are possible but also their relative likelihoods, enabling risk-sensitive decisions (e.g., in finance or safety-critical engineering) that account for the probability mass distribution across all potential outcomes.
Execute fast uncertainty quantification in high dimensions by using the polyhedral partition algorithm. This means that as new test data arrives, the system can rapidly determine a prediction set by looking up precomputed geometric regions (polyhedra), drastically reducing inference time compared to methods that require solving complex optimization problems for every candidate outcome.
Provide exact calibration guarantees in multivariate settings using the randomized CPD/SDOT mechanism. This means that any uncertainty region derived from the system is mathematically guaranteed to have the correct coverage probability, a feat previously unattainable without strong parametric assumptions or asymptotic limits.

Abstract

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in O(n 3) time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.