Exact Rank and Convex Calibration Dimension Lower Bounds for the Multi-Label F1 Loss

Mingyuan Zhang

Exact Rank and Convex Calibration Dimension Lower Bounds for the Multi-Label F1 Loss

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper determines the exact rank of the multi-label F1 loss matrix and proves that every convex calibrated surrogate for it must have quadratic prediction dimension.

desk verdict A rigorous theory paper that nails the exact rank and the first quadratic convex-calibration-dimension lower bound for multi-label F1; it deserves refereeing. read the letter →

arxiv 2608.08399 v1 pith:AMACEV22 submitted 2026-08-09 cs.LG stat.ML

Mingyuan Zhang This is my paper · ORCID

classification cs.LGstat.ML MSC 68T0515A03

keywords multi-labelclassificationF1lossconvexcalibrationdimensionloss-matrixranksubset-incidencematrixCauchyBayes-optimalpredictionsurrogate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper determines the exact linear-algebraic size of the instance-wise $F_1$ loss for multi-label classification with $s$ labels: under the convention $F_1(\varnothing,\varnothing)=1$, the score matrix, the shifted loss matrix, and the unshifted loss matrix all have rank $s^2-s+2$, and the loss columns have affine dimension $s^2-s+1$. It then proves that this quadratic size is not merely an upper bound: every convex calibrated surrogate for the $F_1$ loss must use prediction dimension at least $(s-\lfloor s/3\rfloor)(\lceil\sqrt{s\lfloor s/3\rfloor}\rceil-1)$, which is $(2/(3\sqrt{3})-o(1))s^2$. Together with the affine-dimension upper bound, this gives $\operatorname{CCdim}(L^{F_1})=\Theta(s^2)$, settling whether the earlier quadratic-dimensional convex surrogates could be beaten. A reader should care because $F_1$ is nondecomposable across labels: the result shows that any convex surrogate that preserves Bayes-optimal $F_1$ decisions must pay a quadratic cost in prediction dimension.

What carries the argument

The load-bearing objects are the subset-incidence matrices $W_k$, whose rows are the $k$-subsets and whose columns are labels, and the positive-definite Cauchy matrix $C_{kk'}=2/(k+k')$; the score matrix factors as $K=P(C\otimes I_s)P^{\top}$, letting the rank calculation pass through the incidence matrices. For the calibration lower bound, the central mechanism is a witness distribution on layered outcome sizes whose Bayes-optimal reports are exactly the supersets of a fixed core set, creating tied loss columns whose difference span has dimension $hn$; the feasible-subspace lower bound then converts that local affine dimension into a lower bound applying to every convex calibrated surrogate.

What would settle it

For $s=3$, the rank formula predicts rank $8$ for the $8\times 8$ $F_1$ score matrix; a direct symbolic rank computation returning any other value would refute the rank theorem. For the calibration bound (which gives $hn=2$ at $s=3$), exhibiting any convex calibrated surrogate with prediction dimension $1$ for the three-label $F_1$ loss would falsify the lower bound.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is two-fold. First, the $F_1$ score matrix on nonempty label sets factors as $K=P(C\otimes I_s)P^{\top}$, where $P$ is a block-diagonal matrix of subset-incidence matrices $W_k$ and $C$ is the positive-definite Cauchy matrix with entries $2/(k+k')$; the rank identity $\operatorname{rank}(XMX^{\top})=\operatorname{rank}(X)$ for $M\succ 0$ then forces $\operatorname{rank}(K)=\operatorname{rank}(P)=s^2-s+1$, and the isolated empty-set entry plus a null-space analysis of $L$ give $\operatorname{rank}(F)=\operatorname{rank}(L-J)=\operatorname{rank}(L)=s^2-s+2$ and $\operatorname{affdim}(L)=s^2-s+1$. Second, because exact rank alone cannot lower-bound arbitrary convex surrogates, the paper constructs a witness distribution supported on outcome-cardinality layers $1,\dots,h$ whose Bayes-optimal predictions are exactly all supersets $T\cup C$ of a fixed core set $T$ of size $\lfloor s/3\rfloor$; the active loss columns restricted to the support have affine dimension $hn$, where $n=s-\lfloor s/3\rfloor$ and $h=\lceil\sqrt{s\lfloor s/3\rfloor}\rceil-1$. Applying the feasible-subspace lower bound for convex calibration dimension yields $\operatorname{CCdim}(L^{F_1})\ge hn=(2/(3\sqrt{3})-o(1))s^2$, and with the upper bound this establishes $\Theta(s^2)$.

Load-bearing premise

Everything rests on the imported feasible-subspace lower bound being valid for a witness distribution whose Bayes-optimal predictions are a large tied family (all supersets of a core set) rather than a unique report; if that bound requires a unique minimizer, the quadratic lower bound collapses.

Editorial extensions

If this is right

Any convex calibrated surrogate for the $s$-label $F_1$ loss must use $\Theta(s^2)$ prediction dimensions, so the earlier $s^2+1$-coordinate constructions are essentially optimal in order.
The exact rank $s^2-s+2$ sharpens the previous $\operatorname{rank}(L^{F_1}-J)\le s^2+1$ and yields the affine-dimension upper bound $\operatorname{CCdim}(L^{F_1})\le s^2-s+1$.
Bayes-optimal $F_1$ prediction needs a quadratic number of degrees of freedom in the worst case, matching the quadratic sufficiency of plug-in statistics.
The lower bound is insensitive to the empty-set convention $F_1(\varnothing,\varnothing)=1$ versus $0$, because the witness distribution assigns no mass to the empty outcome.
The constant gap between $(2/(3\sqrt{3})-o(1))s^2$ and $s^2-s+1$ remains open; the paper identifies harder trigger distributions or sub-affine surrogates as the two routes to closing it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The layered-core-set construction is a template: any set-similarity loss whose Bayes-optimal report is a simplex face with large active-column affine dimension should inherit a quadratic calibration-dimension lower bound.
The same factorization with a generalized Cauchy kernel may give exact-rank formulas for the asymmetric $F_\beta$ family, a direction the paper mentions but does not pursue.
A testable extension is a trigger distribution with several interacting core sets, which could enlarge the active face beyond $hn$ and improve the constant factor toward the upper bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a solid theory paper. It settles two open questions: the exact rank of the F1 loss matrix under the stated empty-set convention, and the first quadratic lower bound on convex calibration dimension for F1. I checked the rank proof and the lower-bound construction; both hold up.

The rank part is genuinely elegant. The nonempty score block factors through subset-incidence matrices and a positive-definite Cauchy matrix; the positive-middle rank identity gives the rank, and the all-ones shift is handled with a null-space argument. The affine-dimension count follows cleanly. The step where 1_{N-1} is shown to lie in range(K) is particularly careful.

The lower bound is the real contribution. The paper constructs a witness distribution where all supersets of a fixed core set are Bayes-optimal, shows the active loss columns restricted to the support have affine dimension hn, and then applies the Ramaswamy-Agarwal feasible-subspace bound. The decomposition into label-swap and cardinality-change directions is intricate but internally consistent. The Cauchy nonsingularity, the injectivity of Psi, and the direct sum S ⊕ H all check out on inspection. I found no circularity and no parameter fitting.

The only external load-bearing piece is Theorem 16 of Ramaswamy and Agarwal (2016), imported without proof. The use here matches the stated theorem for arbitrary trigger points, including boundary points with multiple Bayes-optimal reports, so I do not see a problem, but it is the one place a referee should re-read carefully. The self-citation to Zhang et al. (2020) is disclosed and is used for context and upper-bound comparison, not to prop up the new lower bound.

Minor soft spots: the constant in the lower bound is far from the upper bound, so exact CCdim remains open; and the paper is purely mathematical, so there is no empirical component to evaluate. Neither affects the central claims. Also, the rank result depends on the convention F(∅,∅)=1, and the paper transparently reports what changes under the alternative convention.

This paper deserves a serious referee. If the referee confirms the external theorem's applicability, I would accept. I would cite it in my own work and would bring it to a reading group.

Referee Report

0 major / 3 minor

Summary. The paper determines the exact rank and column-affine dimension of the instance-wise F1 loss matrix for multi-label classification with s labels, and proves a quadratic lower bound on the convex calibration dimension. Under the convention F1(∅,∅)=1, Theorem 4.3 shows rank(F)=rank(L−J)=rank(L)=s^2−s+2 and affdim(L)=s^2−s+1; the appendix factors the nonempty score block as P(C⊗I_s)P^T via subset-incidence matrices and a positive-definite Cauchy matrix. Theorem 5.1 constructs a conditional distribution whose Bayes-optimal reports are exactly all supersets of a fixed core set, shows the corresponding active loss columns restricted to the witness support have affine dimension hn with t=⌊s/3⌋, n=s−t, h=⌈√(st)⌉−1, and applies the feasible-subspace lower bound of Ramaswamy and Agarwal (2016) to obtain CCdim(L^{F1})≥(2/(3√3)−o(1))s^2. Combined with CCdim(L^{F1})≤affdim(L), this yields CCdim(L^{F1})=Θ(s^2). Remark 4.5 verifies the alternative empty-set convention.

Significance. This is a substantive contribution. The exact-rank result closes the gap left by the s^2+1 upper bounds of Nowak et al. (2019) and Zhang et al. (2020), and the lower bound is the first F1-specific quadratic lower bound on convex calibration dimension. The lower bound is genuinely Bayes-geometric rather than a corollary of rank, and the proof is carried out in full: the factorization, the injectivity of the coefficient map Ψ, the direct decomposition D=S⊕H, and the feasible-subspace count are all explicit. The construction is parameter-free up to the scale s, and the asymptotic constant is stated. The paper is also careful about the empty-set convention and explicitly acknowledges that the exact constant in the Θ(s^2) bound remains open.

minor comments (3)

[Section 5, Step 4 (Eq. 6)] The feasible-subspace lower bound is applied to a trigger point p that lies on the boundary of the probability simplex and whose Bayes-optimal set is a face rather than a singleton. This is legitimate because Theorem 16 of Ramaswamy and Agarwal (2016) is stated for arbitrary p in the trigger set, but the paper would be easier to read if it said so explicitly in one sentence.
[Section 5, Step 3c] In the definition of the swap space S, the condition Z_{·,O}1_O=0_h is compact; a sentence stating that the optional block of every row lies in W_swap would make the subsequent dimension count more immediately transparent.
[Section 4, Proof of Theorem 4.3, Step 3] In the converse containment for ker L, the fact that 1_+^T z=0 for every z∈ker K is used; this follows from 1_+∈range(K) and symmetry of K, and stating it in the converse sentence would avoid a small ambiguity in the exposition.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the exact-rank and CC-dimension proofs are self-contained, and the only external ingredient is the Ramaswamy–Agarwal feasible-subspace theorem.

full rationale

The paper's derivation chain does not reduce to its inputs. Theorem 4.3 is proved from first principles: the nonempty score matrix is factored as K=P(C⊗I_s)P^T, Lemma 4.1 computes rank(W_k), Lemma 4.2 proves the Cauchy matrix is positive definite, and the positive-middle rank identity gives rank(K)=rank(P). The null-space argument for L and the affine-dimension calculation are explicit and self-contained. Corollary 4.4 then follows directly from Theorem 4.3 and the external affine-dimension upper bound, not from any fitted parameter. For Theorem 5.1, the witness distribution p is constructed with explicit marginals on each cardinality layer, the Bayes-optimal reports are derived as exactly the supersets of T, and the active-column affine dimension hn is computed through the injective map Ψ and the decomposition D=S⊕H. The only external ingredient is Eq. (6), Theorem 16 of Ramaswamy and Agarwal (2016), which is cited as a general convex-calibration lower bound and applied to the constructed p; this is an external mathematical result, not a self-citation. The paper does cite Zhang et al. (2020), which shares an author, but that citation is used for context and previous upper-bound work; the new lower bound does not rely on it, and the exact-rank theorem independently proves the improved rank rather than importing it. There is no fitted-input-called-prediction step, no definitional equivalence, and no self-citation chain that forces the main results. The only correctness risk, if any, would be a hidden hypothesis in the external feasible-subspace theorem when applied to a boundary trigger set with multiple Bayes-optimal reports, but the paper's use matches the theorem's stated form, and this is a correctness concern rather than circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the F1 definition, the standard example-based decision-theoretic setting, two external theorems from Ramaswamy and Agarwal (2016), and standard convex-geometry facts. The only hand-set construction parameter is the core size t; no new physical or model entities are introduced.

free parameters (1)

core set size t = t = floor(s/3)
Design choice in the witness distribution of Theorem 5.1. The lower bound hn depends on t; fixing t = floor(s/3) yields n approximately 2s/3 and h approximately s/sqrt(3) and the stated constant 2/(3 sqrt(3)). It is not fitted to data and the theorem is stated for this choice.

assumptions (6)

domain assumption F1 score convention F(empty, empty) = 1 (Eq. 1)
Defines the loss matrix; the alternative convention changes rank(L) by 1 but leaves the CCdim lower bound unaffected (Remark 4.5).
domain assumption Decision-theoretic, example-based F1 loss is the object of study; micro/macro-averaged F1 variants are excluded (Section 2)
Results are for per-instance loss over label sets; other F1 aggregations have different Bayes analyses and are not covered.
standard math Feasible-subspace lower bound CCdim(L) >= ||p||_0 - mu_{Q_B}(p) - 1 (Theorem 16 of Ramaswamy and Agarwal 2016, Eq. 6)
External theorem used in Step 4 of Theorem 5.1; not proved in this paper.
standard math Affine-dimension upper bound CCdim(L) <= affdim(L) (Theorem 12 of Ramaswamy and Agarwal 2016, Eq. 5)
External theorem used for Corollary 4.4.
standard math Relative-interior points of a finite polytope have positive barycentric coordinates over all vertices
Used in Step 1 of Theorem 5.1 to construct full-support layer distributions; a proof sketch is given in the text.
standard math Subset-incidence matrix rank (Lemma 4.1) and Cauchy-matrix nonsingularity (Lemma 4.2, Appendix A.2, Step 3c)
Proved in the paper; listed for completeness of the audit.

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Cite this review

Pith. "Pith review of Exact Rank and Convex Calibration Dimension Lower Bounds for the Multi-Label F1 Loss." pith.science (2026). https://pith.science/paper/AMACEV22

@misc{pith2026260808399,
  author       = {Pith},
  title        = {Pith review of: Exact Rank and Convex Calibration Dimension Lower Bounds for the Multi-Label F1 Loss},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMACEV22}},
  note         = {Machine review of arXiv:2608.08399}
}

abstract

The instance-wise $F_1$ measure is a central performance measure for multi-label classification. For a problem with $s$ labels, it defines a $2^s\times 2^s$ loss matrix. Previous work exhibited $s^2+1$-coordinate affine and shifted low-rank representations and used them to construct quadratic-dimensional convex calibrated surrogates. We determine the exact rank. Under the convention $F_1(\varnothing,\varnothing)=1$, the $F_1$ score matrix, the shifted loss matrix, and the unshifted loss matrix all have rank $s^2-s+2$, while the column-affine dimension of the loss is $s^2-s+1$. The proof factors the nonempty score matrix through subset-incidence matrices and a positive-definite Cauchy matrix. Exact rank does not, by itself, lower-bound the dimension of an arbitrary convex calibrated surrogate. We therefore analyze the Bayes geometry of $F_1$ directly. We construct a distribution for which precisely all supersets of a fixed core label set are Bayes optimal, and show that the corresponding active loss columns, restricted to the witness support, have affine dimension $hn$, where $n=s-\lfloor s/3\rfloor$ and $h=\lceil(s\lfloor s/3\rfloor)^{1/2}\rceil-1$. Applying the feasible-subspace lower bound for convex calibration dimension gives \[ \operatorname{CCdim}(L^{F_1}) \ge \left(\frac{2}{3\sqrt{3}}-o(1)\right)s^2. \] Together with the quadratic upper bound, this establishes $\operatorname{CCdim}(L^{F_1})=\Theta(s^2)$.

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Works this paper leans on

17 extracted references · 11 canonical work pages · cited by 1 Pith paper

[1]

On consistent surrogate risk minimization and property elicitation

Arpit Agarwal and Shivani Agarwal. On consistent surrogate risk minimization and property elicitation. In Proceedings of the 28th Conference on Learning Theory, volume 40 of PMLR, pages 4--22, 2015

work page 2015
[2]

Dembczy\' n ski, Willem Waegeman, Weiwei Cheng, and Eyke H\"ullermeier

Krzysztof J. Dembczy\' n ski, Willem Waegeman, Weiwei Cheng, and Eyke H\"ullermeier. An exact algorithm for F-measure maximization. In Advances in Neural Information Processing Systems 24, pages 1404--1412, 2011

work page 2011
[3]

Optimizing the F-measure in multi-label classification: Plug-in rule approach versus structured loss minimization

Krzysztof Dembczy\' n ski, Arkadiusz Jachnik, Wojciech Kot owski, Willem Waegeman, and Eyke H\"ullermeier. Optimizing the F-measure in multi-label classification: Plug-in rule approach versus structured loss minimization. In Proceedings of the 30th International Conference on Machine Learning, volume 28 of PMLR, pages 1130--1138, 2013

work page 2013
[4]

Embedding dimension of polyhedral losses

Jessie Finocchiaro, Rafael Frongillo, and Bo Waggoner. Embedding dimension of polyhedral losses. In Proceedings of the 33rd Conference on Learning Theory, volume 125 of PMLR, pages 1558--1585, 2020

work page 2020
[5]

Frongillo, and Bo Waggoner

Jessica Finocchiaro, Rafael M. Frongillo, and Bo Waggoner. Unifying lower bounds on prediction dimension of convex surrogates. In Advances in Neural Information Processing Systems 34, pages 22046--22057, 2021

work page 2021
[6]

Frongillo, and Bo Waggoner

Jessie Finocchiaro, Rafael M. Frongillo, and Bo Waggoner. An embedding framework for the design and analysis of consistent polyhedral surrogates. Journal of Machine Learning Research, 25(63):1--60, 2024

work page 2024
[7]

Rafael Frongillo and Ian A. Kash. On elicitation complexity. In Advances in Neural Information Processing Systems 28, pages 3258--3266, 2015

work page 2015
[8]

Koyejo, Nagarajan Natarajan, Pradeep K

Oluwasanmi O. Koyejo, Nagarajan Natarajan, Pradeep K. Ravikumar, and Inderjit S. Dhillon. Consistent multilabel classification. In Advances in Neural Information Processing Systems 28, pages 3321--3329, 2015

2015

Show all 17 references

[9]

Sharp analysis of learning with discrete losses

Alex Nowak, Francis Bach, and Alessandro Rudi. Sharp analysis of learning with discrete losses. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of PMLR, pages 1920--1929, 2019

1920
[10]

Ramaswamy and Shivani Agarwal

Harish G. Ramaswamy and Shivani Agarwal. Classification calibration dimension for general multiclass losses. In Advances in Neural Information Processing Systems 25, pages 2087--2095, 2012

2012
[11]

Ramaswamy and Shivani Agarwal

Harish G. Ramaswamy and Shivani Agarwal. Convex calibration dimension for multiclass loss matrices. Journal of Machine Learning Research, 17(14):1--45, 2016

2016
[12]

Ramaswamy, Shivani Agarwal, and Ambuj Tewari

Harish G. Ramaswamy, Shivani Agarwal, and Ambuj Tewari. Convex calibrated surrogates for low-rank loss matrices with applications to subset ranking losses. In Advances in Neural Information Processing Systems 26, pages 1475--1483, 2013

2013
[13]

Ramaswamy, Balaji Srinivasan Babu, Shivani Agarwal, and Robert C

Harish G. Ramaswamy, Balaji Srinivasan Babu, Shivani Agarwal, and Robert C. Williamson. On the consistency of output code based learning algorithms for multiclass learning problems. In Proceedings of the 27th Conference on Learning Theory, volume 35 of PMLR, pages 885--902, 2014

2014
[14]

On the Bayes-optimality of F-measure maximizers

Willem Waegeman, Krzysztof Dembczy\' n ski, Arkadiusz Jachnik, Weiwei Cheng, and Eyke H\"ullermeier. On the Bayes-optimality of F-measure maximizers. Journal of Machine Learning Research, 15(103):3513--3568, 2014

2014
[15]

Revisiting F-measure optimization in multi-label classification: A sampling-based approach

Zixun Wang. Revisiting F-measure optimization in multi-label classification: A sampling-based approach. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16845--16854, 2026

2026
[16]

Chai, Wee Sun Lee, and Hai Leong Chieu

Nan Ye, Kian Ming A. Chai, Wee Sun Lee, and Hai Leong Chieu. Optimizing F-measures: A tale of two approaches. In Proceedings of the 29th International Conference on Machine Learning, pages 289--296, 2012

2012
[17]

Ramaswamy, and Shivani Agarwal

Mingyuan Zhang, Harish G. Ramaswamy, and Shivani Agarwal. Convex calibrated surrogates for the multi-label F-measure. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of PMLR, pages 11246--11255, 2020

2020

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

On consistent surrogate risk minimization and property elicitation

Arpit Agarwal and Shivani Agarwal. On consistent surrogate risk minimization and property elicitation. In Proceedings of the 28th Conference on Learning Theory, volume 40 of PMLR, pages 4--22, 2015

work page 2015

[2] [2]

Dembczy\' n ski, Willem Waegeman, Weiwei Cheng, and Eyke H\"ullermeier

Krzysztof J. Dembczy\' n ski, Willem Waegeman, Weiwei Cheng, and Eyke H\"ullermeier. An exact algorithm for F-measure maximization. In Advances in Neural Information Processing Systems 24, pages 1404--1412, 2011

work page 2011

[3] [3]

Optimizing the F-measure in multi-label classification: Plug-in rule approach versus structured loss minimization

Krzysztof Dembczy\' n ski, Arkadiusz Jachnik, Wojciech Kot owski, Willem Waegeman, and Eyke H\"ullermeier. Optimizing the F-measure in multi-label classification: Plug-in rule approach versus structured loss minimization. In Proceedings of the 30th International Conference on Machine Learning, volume 28 of PMLR, pages 1130--1138, 2013

work page 2013

[4] [4]

Embedding dimension of polyhedral losses

Jessie Finocchiaro, Rafael Frongillo, and Bo Waggoner. Embedding dimension of polyhedral losses. In Proceedings of the 33rd Conference on Learning Theory, volume 125 of PMLR, pages 1558--1585, 2020

work page 2020

[5] [5]

Frongillo, and Bo Waggoner

Jessica Finocchiaro, Rafael M. Frongillo, and Bo Waggoner. Unifying lower bounds on prediction dimension of convex surrogates. In Advances in Neural Information Processing Systems 34, pages 22046--22057, 2021

work page 2021

[6] [6]

Frongillo, and Bo Waggoner

Jessie Finocchiaro, Rafael M. Frongillo, and Bo Waggoner. An embedding framework for the design and analysis of consistent polyhedral surrogates. Journal of Machine Learning Research, 25(63):1--60, 2024

work page 2024

[7] [7]

Rafael Frongillo and Ian A. Kash. On elicitation complexity. In Advances in Neural Information Processing Systems 28, pages 3258--3266, 2015

work page 2015

[8] [8]

Koyejo, Nagarajan Natarajan, Pradeep K

Oluwasanmi O. Koyejo, Nagarajan Natarajan, Pradeep K. Ravikumar, and Inderjit S. Dhillon. Consistent multilabel classification. In Advances in Neural Information Processing Systems 28, pages 3321--3329, 2015

2015