On Pixel-Wise Explanations for Non-Linear Classifier Decisions by Layer-Wise Relevance Propagation

Sebastian Bach, Alexander Binder, Grégoire Montavon, Frederick Klauschen, Klaus‐Robert Müller, Wojciech Samek · PLoS ONE · 2015 · machine-learning · 10.1371/journal.pone.0130140 · record bach2015-on-pixel-wise-explanations-for-non
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Other fields define these differently the same word, defined another way in another discipline
prediction 24 fields · black box 3 fields

Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 16 concept tags on this paper, all 16 are grounded in a quoted sentence. All carry the tagger's gloss of what the term means in this paper, which is not something the article can license.
pixel-wise decomposition layer-wise relevance propagation relevance heatmap conservation law black box pixel-flipping segmentation pre-trained classifier non-linearity Taylor reference point Taylor-type decomposition root point sensitivity maps Taylor residuum locally linear relevance conservation between layers greedy layer-at-a-time propagation state of maximal uncertainty prediction conservation of total relevance across layers system-or-tool-building
Method, as described: redistribute the prediction backwards layer by layer under a conservation constraint until it rests on the input pixels

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relation to other workclaim_relationshow this paper stands to another: agrees, contradicts, extends, qualifies
claim typeclaim_typethe shape of the claim: existence, mechanism, method, critique, prediction, null
concept edgeconcept_edgesan asserted relation between two concepts, with the sentence asserting it
conceptconceptsan idea the paper works with, quoted where the paper uses it
content tagcontent_tagsa finding, definition, quantity, statistic or variable named in the paper
contributioncontributionwhat the paper claims to add
controlcontrolsan experiment run to eliminate a rival explanation
limitationlimitationswhat the paper says its evidence cannot support
mechanismmechanismshow the reported effect is said to come about
methodmethodhow the paper produced its evidence
object of studyobject_of_studywhat the paper is about — the thing it investigates
phenomenonphenomenonthe effect or regularity under study
research questionresearch_questionthe question the paper sets itself
sensesensesa distinct use of a term, quoted where the paper fixes it
transfertransfersa method or concept carried in from another field

The paper

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alternatives 3 · claim_relations 2 · claim_type 1 · concept_edges 2 · concepts 16 · content_tags 17 · contribution 1 · controls 3 · limitations 8 · mechanisms 6 · method 1 · object_of_study 1 · phenomenon 1 · research_question 1 · senses 5 · transfers 1

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facettagfieldconf.whose words
object_of_studythe contribution of each individual input pixel to one non-linear classifier's decision on one image
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object_of_studythe paper's words
research_questionhow to explain why a non-linear image classifier reached a particular decision, at the resolution of single pixels
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research_questionthe paper's words
contributionlayer-wise relevance propagation - a heatmapping concept defined as a set of constraints rather than an algorithm - together with a Taylor-type alternative, both applicable to pre-trained Bag-of-Words and neural classifiers
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contributionthe paper's words
claim_typemethod
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claim_typethe paper's words
phenomenonclassifier opacity: a non-linear model gives a decision without saying what in the input produced it
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phenomenonthe paper's words
methodsystem-or-tool-building
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methodthe paper's words
mechanismsthe relevance held by a unit is redistributed onto the units that fed it, and the total is preserved from one layer to the next, so the prediction survives intact all the way down to the pixels
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mechanisms[0]0.95the tagger's words
mechanismsrelevance is computed one layer at a time rather than globally, which is cheap but can strand relevance in a lower layer where it is then distributed randomly
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mechanisms[1]0.95the tagger's words
mechanismsthe gradient at the prediction point points toward the nearest local optimum, which may carry the SAME SIGN as the prediction - so it does not point at the boundary the explanation is relative to
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mechanisms[2]0.95the tagger's words
mechanismsa root is found by taking any candidate with the opposite prediction sign and searching the line between them - the segment must cross zero, so interval intersection locates it
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mechanisms[3]0.95the tagger's words
mechanismsthe root point is specified by CONSTRAINTS rather than by a formula - it must be a root and must lie near the data point, with the rest deferred
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mechanisms[4]0.95the tagger's words
mechanismsthe two approaches differ in what they conserve: the earlier work rectifies activations to keep feature maps non-negative, this one uses SIGNED activations to conserve the relevance quantity across layers
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about this quote: THE QUOTE'S SUBJECT IS A CITATION THE TEXT EXTRACTION DROPPED: the paper's sentence opens with a reference to earlier work, so what 'uses rectified linear units' is that work, which the paragraph compares with this paper's approach - not this paper.
mechanisms[5]0.95the tagger's words
limitationsthe propagation is greedy - computed one layer at a time - and some relevance may hit a dead end in the lower layers and be distributed randomly
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limitations[0]0.95the tagger's words
limitationsthe proposed propagation rules are heuristic: whether relevance proportional to weighted neuron activations can be analytically justified is open
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limitations[1]0.95the tagger's words
limitationsthere is no way to evaluate the quality of a heatmap beyond looking at it
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limitations[2]0.95the tagger's words
limitationsno universal metric for quantifying heatmap quality exists; the paper defers it to future work
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limitations[3]0.95the tagger's words
limitationsa decomposition satisfying the conservation equation is neither unique nor guaranteed to be meaningful - the constraint set does not pin down an answer
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limitations[4]0.95the tagger's words
limitationsthe Taylor approach needs a second point besides the input, and finding it is called a technical difficulty in the paper's own words
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limitations[5]0.95the tagger's words
limitationswhere several minimal-norm roots exist the method must average or integrate over them rather than pick one
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limitations[6]0.9the tagger's words
limitationsa competing method is specific to convolutional architectures, which the paper contrasts with its own range of applicability
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limitations[7]0.9the tagger's words
transfersthe conservation law, taken from physics and imposed on relevance flowing between network layers
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transfers[0]0.85the tagger's words
controlsa quantitative test that can separate two heatmaps which look equally good to a human
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eliminates: two heatmapping methods being ranked on how good they look to a human, which is the judgement the paper exists to replace · outcome: partial
controls[0]0.95the tagger's words
controlsa counterexample testing whether the conservation constraint alone yields a meaningful decomposition
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eliminates: the conservation constraint being credited with producing a meaningful decomposition when it constrains only the total · outcome: failed
controls[1]0.95the tagger's words
controlsapplication to a classifier the authors did not train, to show no retraining is needed
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eliminates: a heatmapping result that depends on the authors having trained the classifier themselves · outcome: held
controls[2]0.95the tagger's words
conceptspixel-wise decomposition
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concepts[0]0.95the paper's words
conceptslayer-wise relevance propagation
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concepts[1]0.95the paper's words
conceptsrelevance
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also in: relevance
concepts[2]0.95the paper's words
conceptsheatmap
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also in: heatmap
concepts[3]0.95the paper's words
conceptsconservation law
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concepts[4]0.95the paper's words
conceptsblack box
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also in: black box
concepts[5]0.95the paper's words
conceptspixel-flipping
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concepts[6]0.95the paper's words
conceptssegmentation
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also in: segmentation
concepts[7]0.95the paper's words
conceptspre-trained classifier
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concepts[8]0.95the paper's words
conceptsnon-linearity
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also in: non-linearity
concepts[9]0.95the paper's words
conceptsTaylor reference point
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concepts[10]0.85the paper's words
conceptsTaylor-type decomposition
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concepts[11]0.95the paper's words
conceptsroot point
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also in: root point
concepts[12]0.95the paper's words
conceptssensitivity maps
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concepts[13]0.95the paper's words
conceptsTaylor residuum
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concepts[14]0.9the paper's words
conceptslocally linear
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concepts[15]0.9the paper's words
senseslayer-wise relevance propagation = whatever satisfies the constraints; several different algorithms qualify
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senses[0]0.95the tagger's words
sensesR_d < 0 = evidence AGAINST the structure being present; R_d > 0 = evidence FOR it
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senses[1]0.95the tagger's words
sensesrelevance at the output layer = the prediction f(x) itself
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senses[2]0.95the tagger's words
sensesheatmap = relevances mapped to a colour space - named, then explicitly put out of scope
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senses[3]0.95the tagger's words
senses'admissible' = satisfying the constraints, which is weaker than 'correct' and the paper says so
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senses[4]0.85the tagger's words
alternativesclaim_type: method
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a second reading reaching the same value. Recorded: method — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,
alternatives.claim_type0.9the tagger's words
alternativesmethod: system-or-tool-building
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a second reading DIFFERING from the recorded value. Recorded: system-or-tool-building — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,
alternatives.method0.8the tagger's words
alternativesphenomenon: which parts of an input image drive a nonlinear classifier's decision
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a second reading DIFFERING from the recorded value. Recorded: classifier opacity: a non-linear model gives a decision without saying what in the input produced it — note: A BLIND second reading by the project on 2026-08-25: the record's own claim type/method/phenomenon were not looked at until every value here was written and substring-verified. Not applied - a person decides. Sampling frame: the 504 staged records carrying no alternatives block, so this never re-measures the 202. All 11 records with 130+ tags were taken as a census and 19 more spread one per discipline across 22 disciplines - the dense non-neuroscience cell is the one the 202 do not contain (inside them, 94-100% of the densest bands are neuroscience; among these 11, one is).,
alternatives.phenomenon0.85the tagger's words
claim_relationsdistinguishes-from → the standard task of optimising a classifier for accuracy and generalisation
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note: the paper's framing move: explaining one decision is ORTHOGONAL to training a good classifier, so a better classifier does not make a better explanation
claim_relations[0]the tagger's words
claim_relationscritiques → the claim that non-linear models' success justifies their opacity
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note: highly successful AND prevented from contributing understanding - the paper puts those in one sentence
claim_relations[1]the tagger's words
concept_edgesnon-linearity —causes→ black box
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direction on the measure: increases · strength of the relation: 0.95 · direction on the measure (stated by the tagger, no quotation): increases Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but endpoint by anaphora: 'This lack of interpretability' = the classifier acting as a black box (previous sentence).
concept_edges[1]the tagger's words
concept_edgesrelevance —part-of→ conservation law
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strength of the relation: 0.85 · note: the conservation law is a property required OF relevance, which is why it constrains rather than computes it Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the formula is read as a conservation law for relevance; part-of is loose.
concept_edges[2]the tagger's words
content_tagsa decomposition satisfying the layer-wise equation is neither unique nor necessarily meaningful
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content_tags.findings[0]0.95the tagger's words
content_tagsa trivially valid decomposition makes every input point the same way as the prediction, which is not a realistic interpretation
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content_tags.findings[1]0.95the tagger's words
content_tagsThe method was applied to a third-party pre-trained ImageNet network without retraining.
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content_tags.findings[2]0.95the tagger's words
content_tagsFurther specification of the relevance propagation rule is deferred to the user or to future work.
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content_tags.findings[3]0.95the tagger's words
content_tagsthere is no direct relationship between a classifier's value at a point and its derivative at the same point, which is the first reason a gradient map is not an explanation
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content_tags.findings[4]0.95the tagger's words
content_tagsDerivatives at the prediction point are not useful for the paper's aim.
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content_tags.findings[5]0.95the tagger's words
content_tagslayer-wise relevance propagation needs no second reference point at all, which is the practical advantage claimed over Taylor decomposition
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content_tags.findings[6]0.95the tagger's words
content_tagsIn the example, the local-gradient explanation gives too large a contribution to an irrelevant direction.
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content_tags.findings[7]0.9the tagger's words
content_tagspixel-wise decomposition
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content_tags.definitions[0]0.95the tagger's words
content_tagslayer-wise relevance propagation
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content_tags.definitions[1]0.95the tagger's words
content_tagsrelevance
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also in: relevance
content_tags.definitions[2]0.95the tagger's words
content_tagsheatmap
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also in: heatmap
content_tags.definitions[3]0.95the tagger's words
content_tagspixel-flipping
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content_tags.definitions[4]0.95the tagger's words
content_tagsthe sign convention that makes a root the state of maximal uncertainty - positive means presence, negative means absence
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content_tags.definitions[5]0.95the tagger's words
content_tagsconservation of total relevance across layers
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content_tags.statistics[0]0.95the tagger's words
content_tagsrelevance
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also in: relevance
content_tags.variables[0]0.95the tagger's words
content_tagsprediction
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also in: prediction
content_tags.variables[1]0.95the tagger's words

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