Demonstration 32

Conformal correlation

A recent paper proposes correlating class membership across conformal prediction sets and reading the result as a map of which classes a classifier confuses. Four experiments, one inside randomized APS exactly as Romano, Sesia and Candès specify it and three on a synthetic classifier with known injected logit perturbations show what the statistic responds to. The close reading is on the critique page.

The question the authors want the matrix to answer is which classes the classifier confuses. The direct answer is the soft confusion matrix,

$$C_{ab} = \mathbb{E}\big[\,p_b \,\big|\, y = a\,\big],$$

the average probability the classifier assigns to class \(b\) over images whose true class is \(a\). It is directional and threshold free, and it needs no calibration set. It does need labeled evaluation data, as any confusion matrix does, and as a conditional mean it summarizes the probabilities rather than determining error rates. The authors never compute it. Every panel below puts the conformal correlation matrix beside the soft confusion matrix on the same data and asks which one answers the question.

We build a classifier whose mistakes we control. There are \(H = 10\) classes. For an image with true class \(y\), the classifier’s logits and reported probabilities are

$$\ell_h = m\,\mathbf{1}[h = y] + G_{yh} + \varepsilon_h, \qquad \varepsilon_h \sim N(0,1) \text{ independently}, \qquad p_h = \frac{e^{\ell_h}}{\sum_{k=1}^{H} e^{\ell_k}}, \qquad h = 1, \dots, H.$$

The margin \(m\) sets the accuracy. The matrix \(G\) sets the confusions: \(G_{yh} > 0\) pushes probability from images of class \(y\) onto class \(h\), and \(G\) need not be symmetric. Panel A uses \(G_{01} = G_{10} = c\) and zero elsewhere. Panel B uses \(G_{01} = \theta c\) and \(G_{10} = (1-\theta) c\). Panel C draws a symmetric \(G\) at random. In every case the injected confusion is known exactly. Pairs with \(G_{ij} = 0\) have no injected confusion, and at the default margin they are separated on almost every image, though not on every image, since the noise is Gaussian.

From this classifier we draw \(n = 1{,}000\) labeled calibration images and 3,000 test images. The sets are the adaptive prediction sets of Romano, Sesia and Candès, which the paper uses. Each calibration image gets the score

$$s = \sum_{k :\, p_k \ge p_y} p_k \;-\; U\, p_y, \qquad U \sim \mathrm{Unif}(0,1),$$

the probability mass at and above the true class, less a random share of the true class’s own mass. The threshold \(q\) is the \(\lceil (n+1)(1-\alpha) \rceil\)-th smallest calibration score. For a test image the classes are taken in decreasing order of \(p_h\) while the mass before them is below \(q\), and the last class taken is kept with probability equal to the fraction of its mass needed to reach \(q\). Marginal coverage is then \(k/(n+1)\) with \(k = \lceil (n+1)(1-\alpha) \rceil\), at least \(1-\alpha\) and within \(1/(n+1)\) of it.

For each test image write \(z_h = 1\) if class \(h\) is in its set and \(0\) otherwise. The conformal correlation matrix is

$$\rho_{ij} = \frac{\mathbb{E}[z_i z_j] - \mathbb{E}[z_i]\,\mathbb{E}[z_j]} {\sqrt{\operatorname{Var}(z_i)\,\operatorname{Var}(z_j)}},$$

with the expectations taken over the 3,000 test images, exactly as the authors define it. Since the \(z_h\) are zeros and ones, each entry is a phi coefficient.

Beside it, each panel computes the soft confusion matrix \(C\) defined above from the same probability vectors. Panel C also uses the symmetric version, the mean pairwise product \(S_{ij} = \mathbb{E}[p_i p_j]\), since the truth there is symmetric. Each panel then changes one thing about \(G\) or the inputs and shows how the two matrices respond.

A. Two classifiers, one law of sets, inside randomized APS

Three classes with equal frequency. Two models assign the probabilities \((0.6, 0.3, 0.1)\) in the orders below, so the true class is always ranked second and both models have accuracy zero. Model 1 labels every \(A\) as \(C\), every \(B\) as \(A\) and every \(C\) as \(B\). Model 2 runs the cycle the other way.

By the score in the setup above, the true label scores \(0.9 - 0.3u\) under both models. The two models are calibrated on the same inputs with the same uniforms, so they get the same threshold, and at test time they see the same inputs and the same uniforms. The panel computes both conformal correlation matrices and both confusion matrices from the resulting sets. In the confusion matrices the row is the true class and the column is the class ranked first.

The two conformal correlation matrices agree up to sampling noise, and the two confusion matrices are transposes of each other. The population laws of the sets are identical: with \(r = (\tau - 0.6)/0.3\), each singleton occurs with probability \((1-r)/3\) and each pair with probability \(r/3\) under either model, so the population matrices are equal exactly. The authors write that the matrix “contains a superset of the information conveyed by the confusion matrix.” Here the sets are produced by randomized APS exactly as Romano, Sesia and Candès specify it, and the matrix does not determine the confusion matrix.

B. The entries are pinned by inclusion rates

Here the confusion term is nonzero for exactly one pair, 0 and 1. The other 44 pairs have no injected confusion. The left heat map is the conformal correlation matrix at the chosen \(\alpha\). The right is the soft confusion matrix, the mean probability the classifier puts on each column class for images of each row class.

In this experiment 44 of the 45 pairs have no injected confusion. Their entries in the conformal correlation matrix are all close to the same negative number, and that number is set by how often each class is included in a set. It says nothing about the classes.

Write \(p_i\) for the fraction of test sets that contain class \(i\). Two classes that never appear in the same set have \(\mathbb{E}[z_i z_j] = 0\), so their entry is

$$\rho_{ij} = \frac{0 - p_i p_j}{\sqrt{p_i(1-p_i)\,p_j(1-p_j)}} = -\sqrt{\frac{p_i\,p_j}{(1-p_i)(1-p_j)}}.$$

With ten balanced classes and singleton sets, \(p_i = 1/10\) for every class and every such entry equals \(-1/9\). The dashed line in the lower plot is this formula evaluated at the observed inclusion rates. It lies on top of the blue line, the average of the 44 entries.

The sliders change the inclusion rates without changing the classifier. Moving \(\alpha\) changes the set sizes, so all 44 entries move together. Adding class imbalance makes common classes appear in more sets than rare ones, so the 44 entries spread out in a pattern that copies the class frequencies. Raising the share of hard inputs, on which the classifier has no margin and so returns a large set, makes all 45 entries positive, including the 44 pairs with no injected confusion.

The authors read a negative entry as dissimilarity and a positive entry as similarity. Here the negative entries are set by the inclusion rates, which here move with \(\alpha\) and the class frequencies while the classifier is unchanged, and the positive entries are produced by input difficulty with no similarity between the classes at all.

C. A symmetric object cannot carry a directional claim

In this experiment a fixed amount of confusion \(c\) is split between two directions. A share \(\theta\) runs from class 0 to class 1, meaning images of class 0 put probability on class 1, and the rest runs from class 1 to class 0. At \(\theta = 1\) images of class 0 are the only ones confused. At \(\theta = 0\) images of class 1 are the only ones confused.

The soft confusion entries \(C_{01}\) and \(C_{10}\) cross as \(\theta\) moves. Read together they say how much confusion there is and which way it runs. The conformal correlation entry \(\rho_{01}\) is one curve, symmetric about \(\theta = 1/2\). Up to sampling noise it takes the same value at \(\theta\) and at \(1 - \theta\), as the readouts show, so it cannot tell a classifier that mistakes 0 for 1 from one that mistakes 1 for 0.

This is forced by the definition. A correlation matrix is symmetric, so it has \(H(H-1)/2\) distinct off diagonal entries. Directional confusion between class pairs has \(H(H-1)\) numbers. The authors nevertheless read from a positive entry that the classifier “is much more likely to misclassify images of class 91 as class 30 than as class 95,” and write that the matrix “contains a superset of the information conveyed by the confusion matrix.” Neither statement can be true of a symmetric matrix.

D. Resolution collapses as the sets shrink

In this experiment every pair of classes has its own confusion strength \(G_{ij} = G_{ji}\), drawn at random once and then held fixed. The question is whether a statistic can recover the ordering of the 45 pairs by injected strength. The left plot shows the conformal correlation entry for each pair against its injected strength, at the chosen \(\alpha\). The middle plot shows the mean pairwise product \(S_{ij} = \mathbb{E}[p_i p_j]\) against the same strengths. The right plot shows the rank correlation between each statistic and the injected strengths, as \(\alpha\) varies.

The probability based statistic recovers the ordering well at every \(\alpha\), and it has no \(\alpha\) to choose. The conformal correlation entries sit at the floor from panel B for most pairs and rise only for the few strongest, so their rank correlation with the injected strengths is lower at every \(\alpha\). It falls further as \(\alpha\) grows, because larger \(\alpha\) means smaller sets, and singleton sets contain no pairwise co-occurrence, which leaves the entries determined by the marginal inclusion rates.

The authors sweep \(\alpha\) over four values and present the dependence on \(\alpha\) as a feature. The dependence is the statistic recovering, four coarse steps at a time, information that the threshold removed and that the probabilities still contain. The randomization in APS adds noise of its own. Untick the box to remove it, and the sets grow instead.

The paper is Perlo, Chiasserini, De Veciana and Malandrino, Computer Communications 247 (2026) 108398. APS sets and scores follow Romano, Sesia and Candès (NeurIPS 2020), randomized as in the original. Untick the box to see the non randomized sets, which are much larger on a confident classifier. Source: js/demo32.js.