CoFilter Method: A Multi-Domain Filter Approach

• CoFilter Method is a multifaceted approach that employs an explicit filtering step to improve decision rules in fields such as statistical multiple testing, image filtering, anomaly detection, and language-model pre-training.
• It adapts filtering principles to reduce conservativeness in hypothesis testing, preserve texture boundaries via co-occurrence statistics, refine cost volumes for anomaly scoring, and select target data using likelihood ratios.
• Its implementations demonstrate practical benefits including increased discovery rates in GWAS, improved edge preservation in images, enhanced AUROC in anomaly detection benchmarks, and efficient data selection for language model pre-training.

Searching arXiv for papers associated with “CoFilter Method” and closely related usages. Search query: "CoFilter arXiv" The expression “CoFilter Method” is polysemous in the arXiv literature. In its most specific usage, CoFilter denotes conditional testing after filtering for multiple testing of partial conjunction hypotheses, introduced as a two-step procedure that removes conservative partial-conjunction $p$-values before multiplicity correction (Dickhaus et al., 2021). In other contexts, the label is also used for the Co-occurrence Filter (CoF), a boundary-preserving image filter (Jevnisek et al., 2017); as an informal shorthand for CostFilter-AD, a cost-volume filtering plug-in for unsupervised anomaly detection (Zhang et al., 2 May 2025); and as an informal shorthand for CoLoR-Filter, a data-selection method for targeted language-model pre-training (Brandfonbrener et al., 2024). A later neuroimaging paper applies the partial-conjunction CoFilter framework to activation localization across subjects or tasks (Dey et al., 5 Aug 2025). This suggests a family resemblance rather than a single method: each variant inserts an explicit filtering stage before a downstream decision rule.

1. Terminological scope and disambiguation

The term is best understood by distinguishing canonical names from secondary usages.

Label in practice	Canonical name in paper	Domain
CoFilter	CoFilter (conditional testing after filtering)	Partial conjunction multiple testing
CoFilter Method	Co-occurrence Filter (CoF)	Boundary-preserving image filtering
CoFilter	CostFilter-AD	Unsupervised anomaly detection
CoFilter	CoLoR-Filter	Targeted LM pre-training

The statistically oriented CoFilter is explicitly named in "A procedure for multiple testing of partial conjunction hypotheses based on a hazard rate inequality" (Dickhaus et al., 2021). The image-processing paper "Co-occurrence Filter" states that CoF is also referred to as the “CoFilter Method” (Jevnisek et al., 2017). The anomaly-detection paper states that CostFilter-AD is abbreviated there as “CoFilter” for cost filtering, while also noting that the official terminology used in the paper is “CostFilter-AD” (Zhang et al., 2 May 2025). The language-model paper states that if one encounters “CoFilter,” it is an informal shorthand for CoLoR-Filter, but that the authors’ canonical name and framing are CoLoR-Filter (Brandfonbrener et al., 2024).

A recurring misconception is therefore terminological: the same label does not identify a single established algorithm across fields. In statistical multiple testing, CoFilter is a formal method name; in the other cases, it is either an alias or a shorthand.

2. Co-occurrence Filter in image processing

In image processing, the Co-occurrence Filter (CoF) is a boundary-preserving, texture-aware image filter inspired by the bilateral filter (BF), but it replaces BF’s range Gaussian with a learned co-occurrence weighting that reflects which intensity or color pairs tend to appear together in the training data (Jevnisek et al., 2017). Its motivation is that BF preserves any large intensity jump, whereas CoF distinguishes edges from boundaries by using local co-occurrence statistics: pixel values that frequently co-occur inside a texture are smoothed together even when their intensities differ substantially, while pixel pairs that rarely co-occur across texture boundaries are not averaged.

The bilateral filter is written as

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$

with

$$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$

and

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$

CoF replaces $w_r$ with a co-occurrence weight $w_c$:

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$

where

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$

For grayscale images, intensities are quantized to $K$ bins, and the spatially weighted co-occurrence counts are

$$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$

With marginal counts

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$0

the paper uses the symmetric normalization

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$1

The paper also lists conditional and symmetrized-conditional alternatives. The limiting cases are instructive: as the spatial kernel bandwidth $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$2, CoF becomes identity; as $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$3, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$4 and CoF reduces to Gaussian smoothing in the image plane.

For color images, direct RGB co-occurrence is infeasible, so the method clusters colors, typically in CIELab, using $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$5-means. Let

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$6

Hard cluster-space co-occurrence is

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$7

and the soft approximation is

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$8

The resulting guided filter is

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$9

The reported practical profile is local rather than non-local: hard co-occurrence collection is $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$0, filtering is $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$1, and the soft approximation adds $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$2, with typical settings Lab color space, $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$3 clusters, and window $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$4. The paper also distinguishes Iterative CoF (I-CoF), which fixes $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$5 across passes and converges quickly while preserving boundaries, from Rolling CoF (R-CoF), which recomputes $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$6 after each pass and yields stronger homogenization at the cost of possible boundary degradation. Relative to BF, Guided Filter, Domain Transform, L0 smoothing, Rolling Guided Filter, semantic filtering, WLS with diffusion distances, and Non-Local Means, CoF is positioned as a method that smooths repeated textures without sacrificing texture boundaries.

3. CoFilter for partial conjunction multiple testing

In statistics, CoFilter is a two-step procedure for multiple testing of partial conjunction (PC) hypotheses, designed to reduce the conservativeness of standard PC $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$7-value testing (Dickhaus et al., 2021). For $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$8 studies and target replicability level $$w_s(\|p-q\|) = \exp\!\left(-\frac{\|p-q\|^2}{2\sigma_s^2}\right),\quad w_r(\Delta I) = \exp\!\left(-\frac{\|\Delta I\|^2}{2\sigma_r^2}\right),$$9, the PC null hypothesis is

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$0

If the ordered study-level $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$1-values are

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$2

the Fisher-based PC statistic combines the $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$3 largest $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$4-values:

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$5

with

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$6

The method’s motivation is that these PC $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$7-values are often conservative under the composite null, so many very large $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$8 contribute to multiplicity but not to discoveries. CoFilter therefore applies a filter threshold $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big).$$9:

$w_r$0

For selected features, it forms conditional PC $w_r$1-values

$w_r$2

and then applies a multiple testing procedure such as Bonferroni, Holm, BH, Storey, or BY on the filtered family.

The theoretical contribution is the validity of these conditional PC $w_r$3-values when the null $w_r$4-values are uniform and Fisher’s combining method is used. The proof is based on a novel inequality in hazard rate order of partial sums of order statistics. The paper states the theorem as follows: if $w_r$5 and $w_r$6 are i.i.d. $w_r$7, and for some $w_r$8, $w_r$9 are any positive random variables, independent of $w_c$0, then

$w_c$1

From this, the paper derives the corollary that

$w_c$2

is a valid conditional $w_c$3-value for $w_c$4.

The method includes a data-adaptive threshold. For BH at level $w_c$5, the threshold at a given $w_c$6 is

$w_c$7

and one may choose

$w_c$8

Under PRDS-type conditions, finite-sample BH control is available; under weak dependence, asymptotic FDP control is given for fixed or random $w_c$9.

The reported application is to multiple GWAS of Crohn’s disease with 8 independent GWAS studies and about 953,241 autosomal SNPs per study. Using Fisher PC $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$0-values for $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$1, CoFilter with fixed $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$2 or greedy $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$3 yielded substantially more discoveries than unfiltered BH or adaptive BH on PC $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$4-values. The paper reports, for $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$5, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$6 for $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$7, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$8 for $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big)\, I(q),$$9, and $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$0 for $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$1.

4. Neuroimaging adaptation of CoFilter

"A New Approach to Partial Conjunction Analysis in Neuroimaging" applies the recently proposed CoFilter method to voxelwise activation localization across subjects or tasks (Dey et al., 5 Aug 2025). For subject $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$2 and voxel $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$3, let $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$4 be the voxelwise $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$5-value for testing inactivity. The number of truly active subjects at voxel $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$6 is

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$7

and the $$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$8-level PC hypothesis is

$$Z_p = \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_c\big(I(p), I(q)\big).$$9

If

$K$0

are the ordered subject-level $K$1-values at voxel $K$2, the paper uses Fisher’s combination on the largest $K$3 ordered $K$4-values:

$K$5

For a threshold $K$6, the filtered set is

$K$7

and the conditional PC $K$8-values are

$K$9

BH is then applied at level $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$0 on the filtered family. The procedure is repeated for all $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$1, and the voxelwise replicability lower bound is

$$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$2

The overall error criterion is

$$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$3

The paper also studies an adaptive or “greedy” threshold over a grid $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$4, choosing the $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$5 that maximizes the number of rejections. It states that Dickhaus et al. (2024) show FDR control for fixed $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$6 for any choice of $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$7, including adaptive choices, when using Fisher’s combination and the conditional scaling $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$8. For varying $$C(i,j) = \sum_{p} \sum_{q \in \mathcal{N}(p)} K_s\big(\|p-q\|\big)\, \mathbf{1}[Q(I(p))=i] \, \mathbf{1}[Q(I(q))=j].$$9, the paper reports strong empirical evidence of overall FDR control, while noting that a full formal proof is not yet provided.

The empirical evaluation includes two simulation regimes and a real dataset. In Simulation 1, with $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$00 subjects, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$01 voxels, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$02 time points, and equi-correlation $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$03, the reported FDR values over 500 repetitions are 0.0262, 0.0256, 0.0234, 0.0191 for CoFilter, compared with 0.0403, 0.0408, 0.0377, 0.0298 for Benjamini–Heller (2009) and 0.0207, 0.0197, 0.0174, 0.0147 for AdaFilter. In Simulation 2, with $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$04 subjects, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$05 voxels in a $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$06 array and SNR $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$07, the reported FDR values are 0.0478, 0.0475, 0.0473, 0.0470 for CoFilter, 0.0477, 0.0474, 0.0472, 0.0469 for Benjamini–Heller (2009), and 0.1066, 0.1152, 0.1142, 0.1141 for AdaFilter.

The real-data study uses the Smeets et al. (2013) food-cue fMRI dataset from OpenNeuro ds000157, comprising 30 healthy women. The paper reports that CoFilter discovers more voxels at very high $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$08 than Benjamini–Heller, makes on average 0.56 more rejections on the full voxel set, differs from Benjamini–Heller in about 18.7\% of voxels, and in about 70\% of those differing voxels CoFilter finds more activation, with about 3 more rejections on average in that subset. Reported Talairach coordinates include $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$09 and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$10 in visual cortex.

5. CostFilter-AD as a cost-filtering “CoFilter” method

In unsupervised anomaly detection, CostFilter-AD introduces explicit matching cost volume filtering into UAD and is abbreviated in the paper as “CoFilter” for cost filtering (Zhang et al., 2 May 2025). The method addresses a failure mode shared by reconstruction-based and embedding-based UAD systems: anomaly maps are typically derived from image-level or feature-level matching, but that matching can be noisy and inaccurate due to imperfect reconstructions, misalignment across views or scales, and the absence of ideal normal templates.

The method constructs a dense matching cost volume from frozen multi-layer features of an input image $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$11 and normal templates $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$12. Using a frozen encoder such as DINO ViT-B/8 or EfficientNet-B4, the paper extracts

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$13

Global cosine matching is then defined by

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$14

and converted to anomaly cost by

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$15

After merging template and spatial candidate indices, the cost volume is reshaped to

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$16

A coarse anomaly map is obtained by global min pooling along the matching dimension:

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$17

Filtering is performed by a 3D U-Net with dual-stream guidance. Spatial guidance (SG) uses multi-layer input features $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$18 to preserve edges and contours, while matching guidance (MG) uses the coarse anomaly map $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$19 to bias the network toward channels likely to contain anomalies. The core Residual Channel-Spatial Attention (RCSA) module concatenates the cost feature with projected guidance,

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$20

then applies residual channel attention

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$21

and residual spatial attention

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$22

After filtering, the final anomaly map is

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$23

and the image-level anomaly score is the average of the top-$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$24 pixels with $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$25.

The paper emphasizes that CostFilter-AD is a generic post-processing plug-in that can be attached to both reconstruction-based and embedding-based pipelines. In reconstruction-based settings, templates can include intermediate denoising steps

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$26

where $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$27 is the frozen diffusion noise predictor. In embedding-based settings, a small number $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$28 of normal images per category is sampled instead of using large memory banks. The filtering network is trained on synthetic anomalies under the GLAD protocol with

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$29

using $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$30 and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$31 in the class-aware focal adaptor.

The implementation details reported are $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$32 feature layers, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$33 for GLAD and AnomalDF, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$34 for HVQ-Trans, input sizes $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$35 for GLAD and AnomalDF and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$36 for HVQ-Trans, and Adam, lr $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$37, batch size 8, 40 epochs, with ReduceLROnPlateau. On A100 40GB, the added runtime and memory are reported as +0.37 s and +2.07 GB for GLAD, +0.07 s and +0.94 GB for HVQ-Trans, and +0.32 s and +0.82 GB for AnomalDF.

The reported gains are benchmark-specific but consistently positive. On MVTec-AD multi-class UAD, mean image/pixel AUROC improves from 97.5/97.3 to 98.7/98.2 for GLAD, from 98.0/97.3 to 99.0/98.0 for HVQ-Trans, and from 96.8/98.1 to 98.5/98.8 for AnomalDF. On VisA, the corresponding changes are 90.1/97.4 to 93.2/98.1, 91.3/98.5 to 93.4/98.6, and 90.5/97.5 to 94.3/99.2. The paper also reports improvements for a unified single-class model and ablations showing that mapping $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$38 to channels is superior to mapping $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$39 to depth, that multiple denoised templates outperform the final-step template alone, and that dual-stream attention plus the full loss gives the best results.

6. CoLoR-Filter as an informal “CoFilter” in language-model pre-training

In language-model pre-training, CoLoR-Filter stands for Conditional Loss Reduction Filtering. The paper states that if one encounters “CoFilter,” it is an informal shorthand for the same method, but that the canonical name is CoLoR-Filter because the method is defined by the conditional reduction in loss of a target-domain model relative to a background model (Brandfonbrener et al., 2024).

The setting is targeted data selection. Let $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$40 be a large corpus, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$41 a small sample from the downstream target distribution, and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$42 a prior dataset, typically sampled from the same distribution as $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$43. For an autoregressive LLM,

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$44

The paper derives an empirical Bayes-inspired objective and approximates it using two small auxiliary models: a marginal model $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$45 trained on $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$46, and a conditional model $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$47 obtained by fine-tuning $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$48 on $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$49. The per-example score is

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$50

Selection is by ranking: choose the bottom-$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$51 examples by $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$52, equivalently those with the largest likelihood ratio $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$53.

The procedure is offline and largely parallelizable. One trains $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$54 on $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$55, fine-tunes it on $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$56 to obtain $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$57, samples a candidate pool $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$58 of size $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$59, computes per-token losses under both auxiliary models on fixed-length 512-token chunks, scores each example by $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$60, then trains the target model on the selected subset. The auxiliary models are reported as $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$61M non-embedding parameters, with 12 layers, 1024 hidden size, 4096 MLP, 64 head dimension, RoPE positional encoding, context length 512, mixed precision (bfloat16), Adam, batch size 256, learning rate $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$62, warmup and cosine decay, and z-loss $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$63. The target models are 1.2B non-embedding parameters, with 24 layers, 2048 hidden size, and 8192 MLP.

The paper gives total compute cost, in model-forwards per token, as

$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$64

where $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$65, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$66, and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$67 for 1.2B vs 150M in the reported setup. The term $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$68 is fully parallelizable.

Two empirical settings are emphasized. For domain adaptation to Books, CoLoR-Filter with a pair of 150M auxiliary models can train a 1.2B target model to match a 1.2B model trained on 25B randomly selected tokens using about 1.5B filtered tokens, or about 25$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$69 less data. The paper also reports a compute reduction of over $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$70 in the example calculation with $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$71B, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$72B, $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$73, and $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$74, comparing 82 to 412.5 units. For a suite of downstream multiple-choice QA tasks, CoLoR-Filter data selected with 150M auxiliary models can train a 1.2B target to match the performance of a 25B random-token baseline with about 3B filtered tokens, or about 11$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$75 less data.

For 150M targets at $$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$76 and 3.1B selected tokens, average accuracy on the multiple-choice QA suite improves from 46.3 for Random 1$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$77 to 49.1 for CoLoR-Filter, exceeding the Random 8$$I'(p) = \frac{1}{Z_p} \sum_{q \in \mathcal{N}(p)} w_s(\|p-q\|)\, w_r\big(I(p)-I(q)\big)\, I(q),$$78 baseline at 47.1. CoLoR-Filter also outperforms Conditional-only (46.0), RHO-down (46.1), RHO-down+prior (48.5), and DSIR (48.3) on average. The paper additionally notes that CoLoR-Filter is most effective when targeting a downstream distribution different from the pretraining source; using it to improve IID losses on C4 does not help.

Across these uses, the label CoFilter consistently refers to a design pattern in which a downstream inference stage is preceded by an explicit filtering operation: co-occurrence weighting before local smoothing, thresholded conditional testing before multiplicity correction, guided cost-volume denoising before anomaly scoring, or likelihood-ratio ranking before target pre-training. The methods are not interchangeable, but they share a common structural principle: filtering is treated as part of the estimator rather than as an incidental preprocessing step.